You are correct in general, although there is a weird exception in the law.
If your parent is your household employee, you are not required to withhold or pay taxes unless:
1. they are providing care for your minor child or adult disabled child,
AND
2. your marital status is either divorced and not remarried; or widow/widower; or your spouse is disabled and unable to care for themself.
In other words, a parent who is single and never married, or parents who are married, can pay their parent for child care and it is NOT subject to household employee tax, but parents who are divorced or widowed and not remarried ARE subject to household employee tax. I don't know why the law is written this way.
https://www.irs.gov/publications/p926#en_US_2024_publink100086740
Assuming that you are in a situation where your parent is not subject to household employee tax (or social security and medicare withholding), you have the option of issuing them a W-2 with the wages in box 1 and boxes 2-6 blank. But a W-2 is not required. If you don't issue a W-2, they report household employee wages not on a w-2, using line 1b of form 1040.
