How do I prevent a hospital across the country from kicking dad out? He has Parkinson’s gait, gout and dementia.

Your father will likely have to go into care while your Mom is herself battling stage IV cancer. Stage IV for pancreatic can be devastatingly fast and I hope Mom's POA has discussed her prognosis. It is unusual to survive long with that disease. I am so very very sorry. Each patient is unique and you do occ. hear of people having more time through meds, but that is unusual. And your Mom will in no wise be able to care for your Dad again. Sadly this does mean placement.
If Dad is in hospital now is the time to enlist the Social Worker at the hospital to find emergency guardianship and placement for your sister or for you and to find placement that would be almost impossible for you to find on your own. They may be able to "buy time" by fudging the facts and claiming he needs rehab, transferring him for a week or so of that, but the fact is he needs 24/7 care. You have no way of knowing this but the facts are he has no conditions that require ACUTE care, which is the only way anyone stays in hospital for any appreciable time. They now send people home next day after mastectomy, gallbladder removal and so on. Acute needs means he has something ONLY a hospital can cure and that is CURABLE. Say acute pneumonia.
I am so sorry. What a blow for your family. I am so sorry.

I'm confused:

1.   Your father is staying with another daughter.

2.    The title of your post addresses how to prevent him from being "kicked" out of a hospital.  How are the two related?  Is he in a hospital in your sister's area?  

This isn't clear and it would make a difference if he's hospitalized and needs to remain so, vs. living with a sister.

3.   You asked how to get your father home "w/o being kicked out".   Kicked out from what?  From where?

It's hard to respond to inconsistent statements; could you please clarify?

Hospitals are not under the "unsafe discharge" law according to my RN daughter. Medicare says when its time to discharge.

You need to call the hospital and tell them the circumstances. But don't be surprised if they tell u not much they can do. You may just need to go get him. And I too wonder why if you have POA you need to set up guardianship. Maybe time to place Dad. Medicaid allows assets to be split. Dads going towards his care, when spent down, Medicaid is applied for. Mom is the Community Spouse and stays in the home and gets enough of their monthly income to live on. Your eventually going to need to place him anyway with you needing to work.

If you're his PoA why do you also need guardianship? Is someone contesting the PoA?

Does the hospital know he has a diagnosis of dementia? If it's not obvious the discharge staff may take him at his word if he says he has a place to go or can get himself "home". Does the hospital know you're his PoA?

If there's nowhere for him to go and the hosp knows it, it's on them to make arrangements.

Sounds like you're being pressured to take him because relative and/or POA. All you have to tell them is unsafe discharge until you can go get him by X date. If that's at a NH, you'll know.