I am the POA for my father. He had a caregiver for about two years which was working out fine. In the last few months he has needed extra help (bathing, etc.) so the doctor recommended a new caregiver who comes several times a week. From the day the caregiver started there have been a number of issues. Money has gone missing from my fathers apartment, she left the stove on and he burned himself, and there have been a number of other issues. Last night after a new issue arose (long story) I decided to fire her. It wasn't something that I took lightly but I believed that it was in his best interest not to allow this woman back into his apartment. I called my sister to let her know, and she got into a huge argument with me. I am the POA for my father, and she told me I don't have the authority to fire her. I am not sure why she she wouldn't be on my side, but the entire situation began an epic fight. Do I have the power to fire a caregiver if I believe she is not suited for my father's care?
I can't understand why your sister would want to keep a bad caregiver though.
Just... Things going missing. The stove left on. These would be characteristic behaviours in a person with dementia: are you absolutely certain that this caregiver is responsible? Also, consider that the upset of having a new person around the place could have been a setback for your father's mental state - it wouldn't be difficult to see how he might be more confused until he got properly used to her.
Try not to let it worry you. The stress of caregiving, and sibling relationships, and wielding POA are all notorious for creating conflict out of nowhere, and out of all proportion to the subject at hand. I hope you'll be able to kiss and make up; but in any case once you've got the facts clear stick to your guns.
Now this can be tricky, if your Dad has Alzheimer's/Dementia there are times when the patient makes up stories. Maybe it was Dad saying these things so he can get back the original caregiver.