Does the state pay the caregiver of a family member?

https://www.agingcare.com/articles/how-to-get-paid-for-being-a-caregiver-135476.htm

But, if there are resources you could get a caregiver agreement that parent will pay you. Then they become an employer and must deduct taxes, FICA, disability, whatever to be in compliance with state law.

No, unless they are eligible for Medicaid or a Veteran. In those cases call the VA ask about Aid and Attendance, or call Medicaid. The payment is only a small stipend.