Is there anything I can do when dad continues to tell people, like social services and doctors, lies about me?

Threaten to resign? Tell social services that the caregiving relationship is no longer tenable? Tell elder care services and your family you can no longer be put in this position. I don't mean to sound flip, but maybe you should meet with APS yourself, tell them the situation and see what they advise.

Get a letter from the doctor stating you are the caregiver to both parents, both with dementia. Keep it handy, more than one copy, and in a safe place.

As difficult as it is you ignore what he is saying.
Sooner or later people will realize that something is amiss with your dad.
You continue to care for mom, you start taking care of dad.
BUT taking care of 1 person with dementia is difficult enough but taking care of 2 is a daunting task. You will need help. Your parents need to pay for caregivers that can help or you need to begin to look for Memory Care for both of them. Please think of your and their safety in your decision, if it becomes unsafe for YOU to care for mom and or dad or if it becomes unsafe for THEM for you to care for them you have to consider placing one or both.

From your profile: "My husband and I have just recently packed and moved my parents from Norton, VA to their new home, directly behind my house, in Evington, VA. My mother has moderate dementia and breast cancer. My dad currently has mild dementia. I am new at being a caregiver. My husband and I currently live in our own home, and I am spending much time at their home preparing meals, taking them to their doctor appointments, and preparing their pills. I know that as the disease progresses, I will need to move into their basement to better care for them."

Are you really willing to move into their basement? What is their financial situation? I sure hope you aren't sacrificing everything in order to provide your siblings with an inheritance!

In regards to the lies, my mother did that, too. At the NH she had written a letter that the aides saw accusing me of trying to get ownership of her condo. Sure, the standard advice is to ignore it, BUT this was just one of many attacks on my mental wellbeing. At least by this point I COULD just shrug it off, since I was being paid $20/hour for any time I spent with my mother by one of the POA brothers.

And, regarding finances again, are your parents contributing anything at all? They live in a house on your land. Do they contribute towards utilities? Food? Anything? If not, why not?

What's going to happen when their care becomes too much for one person (you)? Could they qualify for Medicaid?

Every adult pretty much the world over knows that your name could not have gone on a bank account unless the account holder went to the bank and did it.
The DMV does not let you decide who loses their driver's licenses and who doesn't. A doctor sent a letter stating your father has dementia and is no longer fit to drive. That's why his license isn't valid. The guns being taken away, well that's just plain old common sense. Your mother had dementia first and should never have been living in a house with firearms.
Also correct your father if he tells a harmful lie in your presence. Like saying you stole his money. Speak plainly and simply say, 'that is a lie' and let that be the end of it. Don't engage.
It would also be a good idea to bring in some homecare for them a few hours a week. A hired caregiver to give mom a shower, housekeeping, and to bring them to some of their doctor's appointments.
You'll probably be met with resistance but reassure them by making yourself clear that either they let the caregiver help or they will be put into a NH or MC. Or your siblings can start stepping up and make hired homecare help unnecessary.
You are the one in charge here. No level of showtiming from your father or BS from your siblings will change that.

Everyone knows this. Your siblings know it too. If they are going to visit and carry on about how nothing is wrong with dad, tell them they will no longer be allowed to visit them on your property. That at any time you have the full legal authority to put both of them into a care facility without explanation to siblings or anyone else. They do nothing to help with the caregiving so you will be the one who decides the terms of how they will visit and what they will say to your parents.
That is your call not theirs. Remember it.

You should never had been put on the car title. This makes you responsible for anything involving the car. If Dad is driving and he has an accident killing someone you can be held liable too. You could lose everything because car insurance only covers so much.
Dad assigned you POA for him, do you also have Moms?

What I would do is get rid of the car. As POA you should have the ability to sell it. Out of sight out of mind. Call DMV and see if you need Dad to sign the title with you having POA. Put the proceeds in the bank for them. If still financed maybe u can make a deal with the finance company.

I think you are going to find that you cannot care for two people both having a Dementia. Actually, they probably should not be alone. Dad is not capable if caring for Mom. If I had to pick which one would go to at least to an AL is Dad. He is going to be the one to burn you out. But really, they both should be in an AL. If not now eventually and you need to prepare yourself for that. If you are getting criticized by siblings, even after giving them a copy of the Drs letter, then tell them they can care for Dad.

I think you have taken on more than you have bargained for.

". I took them to see Matt Canning, who told me about your program. I am eager to learn more about your resources."

I think you may have the wrong impression of us. Right now you are on the forum. A forum of Caregivers. Some, like me, our Caregiving is over but we remain in the forum to help others. We are primarily from all over the US, Canada, the UK and other Countries. We are part of "Finding a Place for Mom". I wouldn't consider us a program.

We are not an employment agency. As the forum we can point you in the right direction but cannot pinpoint exactly what resources your area has. Medicaid will be talked about a lot, but every State Administers their Medicaid a little different. Basically the same but with some differences. There are articles you can read. Search a certain problem, former posts relating to it will pop up. Want to search what Assisted Livings are near you, theres a way to search that. Having a certain problem with Mom or Dad, someone on the forum has had the same problem. You want to Vent, great place to do it, just tell us in advance its a vent. Like any forum you take the good with the bad. You take what you need and leave the rest. Sometimes you just need to ignore.

This sounds all too familiar. Luckily, you have a POA. I can only speak from my ow experience but psychiatrists and their team, be they psych. nurses or Social Services have no interest in ascertaining the teuth in these situations. If Mom or Dad are telling lies, this is a normal part of dementia which is usually aimed at the actual person(s) providing care. If a sibling then comes out of the woodwork, so to speak, and starts agreeing with these lies and even encouraging same, this can be devastating, especially as it seems to reinforce the lies and make your position untenable and impossible to defend. There are bad people out there, even in close families, just know that you are doing your best and those that truly know you know that.

Your Dad is diagnosed. And you should be on his account, but you should be on it as POA, and sign all checks his name, with you as his POA. I hope you have had advice of an elder care attorney (also paid for by Dad's estate) to let you know your fiduciary duties under the law, and how to keep meticulous records.
Anyone familiar with dementia will know where all this comes from. Anyone who doesn't should be given web sites so that he or she can get an educations.
And on you go.
I think this gets put in the file titled "No good deed goes unpunished". ;-)

What dad is doing is known as Showtiming; acting like he's fine and fooling others, including doctors, into thinking he's okay & there's nothing wrong with him. He's able to make small talk, and THAT is what convinces people he's fine.........if they were to ask him in depth questions, however, THAT is when they'd discover his deficits! My mother was The Queen of Showtiming, even late into her dementia, b/c she could make small talk and schmooze the ear off a brass monkey! She got turned down for hospice the first time due to Showtiming, when she had 3 months to live! She did get approved a month later, only b/c I literally BEGGED the nurse to take a VERY close look at her and to ignore the schmooze fest and delve deeper than the surface! Fortunately, during that evaluation, mom was Sundowning something fierce (it was later in the afternoon) and was LIVID with the nurse b/c she wouldn't 'take her home' but brought her to her room instead! So she got approved and passed away 2 months and 1 day later. So yeah, Showtiming is a very real thing and these elders get other's believing THEIR stories and making US look like the Bad Guy in the process! You can have cards printed up that say something to the effect of "My father/mother has Alzheimer's so please be patient with their storytelling." Whatever suits the situation.

That said, you are likely in over your head trying to care for TWO parents with dementia/Alz. Moving into their 'basement' (in a modular?) would be an unwise idea, in my opinion, b/c as dementia progresses, the nightmare unfolds (normally) into something unmanageable for one person to handle. With TWO elders suffering from dementia, you're in over your head, my friend. Think about Plan B and also Plan C to manage your folks. Your siblings either need to support you or keep their mouths shut! Telling dad there's nothing wrong with him just feeds the problem and ramps up the Showtiming efforts on dad's part which undermines YOU.

Education is critical, for you and for your siblings. I suggest you read this 33 page booklet (which is a free download) which has THE best information ever about managing dementia and what to expect with an elder who's been diagnosed with it.

Understanding the Dementia Experience, by Jennifer Ghent-Fuller 
https://www.smashwords.com/books/view/210580

My mother was living in a Memory Care Assisted Living facility for just under 3 years when she became wheelchair bound with moderate dementia. She had been living in the regular AL building before that for 4 years, but needed to move into Memory Care when they could no longer manage her needs. You may want to look into MC for your folks as well; they may be able to live together, which would be good. When I worked in a Memory Care ALF in 2019, we had a couple who were physicians who lived together in a large unit there and did quite well as they weren't separated. Their daughter would come by regularly to visit them.

I had POA for both of my parents, and I was on their checking account so I could write checks and handle all of their finances too. I was also the Federal Fiduciary for their VA Aid & Attendance benefits they received; dad had been in the army during WWII and so he was entitled to benefits, which mom continued to get after he died. I'd get RID of dad's car if I were you, as mentioned in other comments here.

Wishing you the best of luck with all you have on your plate. It's a lot; you're going to need help; realize that and formulate your plans to get it, using your folks' money to either hire help or get them placed when their care becomes too much for you to handle.

"Dad did not trust my 2 siblings who have done nothing to help them. Just continually asking for money."

And you took his word for that, did you?

Never mind, it doesn't matter now.

Who is the "everyone" that your father is telling about you? Does it include anyone who a) doesn't know better and b) might make a difference to anything?

For clarity: "showtiming" (see also "sundowning") is a particular, specific phenomenon characteristic of certain types of dementia. It is a response to the stimulus of special occasions and conversations, when the person in question summons the energy to focus and perform better than normal.

What your father is doing is more like confabulation - he wants to talk and it all comes out in a stream of consciousness which includes everything he brings to mind at the moment and mixes together in a mush of words. Like the lady I was with yesterday, who, when she stated she'd already had her lunch and I asked her what she'd eaten, basically just made a menu up on the spot but - clue - included the packet of crackers she spotted on the counter as she was talking.

So your Dad is asked by the nice social worker about who helps him with a, b, and c, and at the same time you get mentioned, and in some (good) moods he'll say that you do everything for them right down to mending the roof, and in other (less good) moods he'll say you do nothing at all and have intentionally left them to starve so that you can take their money. It will also depend on who last said what to him, what's bothering him in the moment, and on how any given question is phrased. It's alarmingly easy to get the answer you're expecting from someone with mental frailty.

Your parents' social workers and doctors know the score. They may (they should) continue to ask questions to ensure that all is well but they're not going to have you arrested for neglect or financial abuse purely on your Dad's say-so.

This must be extremely hurtful and depressing for you; but are there any practical challenges you're having to deal with as a result?

As for the siblings - how did you get on as a family before all this began?