How can a family member be compensated for caring for a parent?

Every state in the US has some sort of in home support services program however only 12 or 13 have programs through Medicaid. There are over 400 programs that provide assistance for care in the US. You are in Texas, which has several programs that are state and federally funded. Both you and your mom have to be eligible though and there are income requirements . You can read about the programs here https://www.google.com/amp/s/www.payingforseniorcare.com/paid-caregiver/texas.html%3ftmpl=amp

Start by searching this site and reading information already available.

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

The usual answer is no, it has to be paid by a care agreement with the person being cared for. But do please look up the recommended site, as I am sure you want the details.

I think you might be able to get a job as a nurse aide ( not certified) or just a “caregiver “ position at any Homecare agency that accepts Medicare and Medicade and ask to take care of your own. I have heard of this before. I have also heard that you can apply for help as a caregiver if your parent is a veteran.
Call one of your local Homecare agencies and see what they have to say it certainly can’t hurt.