How do you obtain guardianship over a parent that has POA?

I would contact an attorney where your dad is right now. If your dad is still competent, he can appoint a new Durable POA and Healthcare POA, as Rainmom said above. Your sister has no say so over that. He can appoint whoever he wants to appoint, including you or another sibling. Let the attorneys handle letting sister know of the change. In some states, the POA's have to be filed with the court or Register of Deeds. The attorney will handle all of that.

If he is not competent to appoint a new DPOA, then I would ask the attorney about filing for Guardian. Most states have a procedure for emergency situations. The suitability of the Guardian will then be determined and you question if your sister is appropriate, assuming she wants to do it.

I would discuss the duties and responsibilities of being Guardian. It's a huge responsibility.

My sister which has POA paper work dropped him off in California with my other siblings. They tried to care for him the best they could, a few weeks later my dad was hospitalized and then sent the nursing home in California. My siblings in California took care any paperwork or decisions on behalf of my dad. They never formally obtained any POA or guardianship papers. I'm not sure why the nursing home in California didn't ask for any paperwork. My siblings in California signed off and released my dad to me which we flew back to Illinois. My sister here doesn't have his best interest at heart. All my other siblings agree with me but my sister is not willing to talk about any other options for care. She is overly obese(400 pounds) and disabled which she is no position to care for him in an apartment. He's currently hospitalized, I wish there was something I can do immediately.

I'm really surprised the NH in CA let you discharge him as you don't have POA?

If your father is still deemed competent he can appoint a new POA through an attorney. If your sister challenges this she will need to go to court to prove him incompetent at the time the new POA was drawn up. This can be done through previous medical records/diagnosis and/or a judge will order new evaluations, usually preformed by two independent doctors. At this point - having taken things this far; court - rather than POA if your father is deemed incompetent your sister will be awarded guardianship which largely is the same as or trumps POA. Conversely you can do the same by going to court and asking for guardianship. I understand that attempting to get guardianship can be a lengthy and expensive process as you need to hire an attorney. I have also been told that frequently in the case of inter-family squabbling a judge will appoint an independent third party guardian who is paid for their services from the person in questions funds. Needless to say it would behoove you and your sister to work this out amicably between yourselves.