Financial responsibility?

Your parents assets need to be split. His going to his care in the NH. Her's taking care of her. See an Elder lawyer.

If he is on Medicaid, Mom is considered the Community spouse. She should be getting enough money to live on from their monthly income. Partial or even all of it.

If he is paying privately, Mom is entitled to be supported with their marital assets.

You may want to get advice from an Elder Law attorney about how to handle the finances in such a situation. Are your parents still happily married despite the separate living, or is there contention between them? Did they combine their finances, or keep separate accounts? You may also need to be aware of Medicaid rules and how those might affect each parent. I wish you well in dealing with this responsibility.

Yes, unless there has been a division of finances.
You as POA should be seeing an elder law attorney as regards all of this.
Your POA sees to payment from your Dad for the expert care you require in managing his finances for his needs.
It seems to me if Dad is in care and self-pay, and mom in home and with her own funds to pay bills, then this division of finances is already DONE?
If so do check in with the attorney. Hope you will update with what you learn.