If I had POA and the person dies and is broke, am I responsible for the funeral bills?

In my County, the coroner's office offers very low cost cremation for persons at need. Check the web site in your County, there met even be a zero cost County alternative.

What if there's no money left and the parent wants to be cremated? Who pays for that?

As POA, your responsibility ends with the person's death. You are not responsible for the person's funeral bills UNLESS, you decided to have a funeral and YOU have made the arrangements and accepted responsibility for payment.

If there is no money in the estate and you wish to have a funeral, it is up to you to figure out how to pay for it. If there is money, the estate can pay but you need to coordinate with the executor on just how much money can be spent.