Is it wrong for a caregiver to pay themselves with their parent's money?

It’s not wrong. You should be compensated for your time and effort.

Just keep records of everything for yourself and your mom so there won’t be any negative repercussions in case Medicaid is needed later for your mom.

If you get paid by your parents, this makes you their employee and thus, as was stated by the other commenters, you need to have a written agreement, do the taxes correctly, etc. Unless your parents have a lot of financial resources for their ongoing care, you do not want to endanger their ability to qualify for Medicaid. Do not co-mingle any of your money with theirs, and lots of other pitfalls to be aware of. In some states the "lookback" period of the application is 5 years. I agree that consulting an elder law attorney who is familiar with estate planning and versed in the Medicaid application would be a wise investment of your parents funds.

If the parent agrees to pay for the care, no, it isn't wrong. But should be done carefully with care plan, with lists of fee for service, rental cost and so on, and the caregiver must use his or her CPA to report those payments that must be reported to the IRS.
There are many repercussions to doing this wrong for both the parent and yourself, so it is time to pay for an hour of Elder Care to get good advice.

Loaded question. No, it is not wrong if setup correctly with a caregiver agreement with taxes, etc withheld and parents are aware and agree. Otherwise you would cause parents problems qualifying for Medicaid if they ever need it.