How responsible will I be for my parents' debts?

Not at all responsible unless you signed for their debt. Anything you co-signed you are responsible for. Keep your name off their accounts so things don't get confused, unless as a POA. They are responsible for their debt when living and their estate is responsible to pay off their debts after their death. Creditors can get liens and etc. on home, care, what have you after death.

Belated thanks to everyone! I'm finding even more debt ... it kind of makes my blood run cold when I see more ...

I would check out how checks should be signed when you are a POA. I was on my Moms accounts. So I signed my name followed by POA. When I signed anything of importance, I asked how they wanted me to sign. Some papers have a "representative" line. This I signed with my name and POA following. Some places wanted me to sign Moms name with mine underneath followed by POA. Always sign with POA following.

You are not responsible for parents debts. If they have no money, they have no money. Children are not responsible and holding a POA ur not responsible. Just be careful what you sign. Do not make yourself responsible for anything.

Your father is responsible for his debts and upon his death his assets in his estate will be used to pay these debts; this will be the responsibility of the executor of his will or the person assigned by the courts to settle his estate in the absence of a will.
HOWEVER some people don't understand POA and don't understand that it is crucial that your money and name not be melded on accounts as anything but his POA or the POD of his accounts. That is to say take care and understand how a poa works. When you sign checks they should be signed with his name. For instance My Dad, by His Son under POA or POA in fact. Never sign your name to anything. Never discuss anything with bill collectors.
Were I you, unless you understand POA very well, I would pay for an hour of time with an attorney to understand how not to get yourself assigned responsibility.

You are if you are joint on anything with them, like a credit card, mortgage, loan, etc. My husband was financial PoA for his mom. She was married to a man with Parkinsons and very stubborn and secretive. We stepped in when they were spiraling out of control from cognitive decline and found they were deeply in debt and barely had enough left of their ss checks every month to buy food. They had a ballooning second mortgage on a home they'd never live long enough to own. We wrote a letter to the mortgage holder explaining the situation and began only paying a fraction of their monthly payments so that it wouldn't trigger the foreclosure process yet. We marked collection letters "Not at this address" and put them back in the mailbox unopened and blocked collection callers. We cancelled any unnecessary services (like the water softener, newspaper, auto insurance so they couldn't drive, etc.) We applied for Medicaid for MIL (her husband wouldn't sign the application). Long story short when they were both finally placed in facilities we locked their home and dropped off the keys at the mortgage holders. The end. So, please don't worry as you manage your parents' financial mess. It is sad that so many won't see the money they are owed but there is nothing that can be done (and those businesses probably have loss insurance that will cover some of it). If your father has diagnosed dementia you cannot be made his PoA, but you can for your mom. I wish you wisdom and peace in your heart as you make the best of your abilities to help them.

No. Unless you sign accepting responsibility for their debt, you are not personally responsible for your parents debts.

POA only gives you the authority to act on their behalf, based on what they would do themselves if they were of sound mind. It does not make you responsible for their past actions. If they have the money to make monthly payments towards the debt you should do that.