Everything is here.
https://www.irs.gov/publications/p926
Briefly, you do not withhold or pay household employee tax if the household employee is your parent, except in one special set of circumstances:
a. your parent is caring for your child under age 18, or your child over age 18 who disabled and unable to care for themselves, AND
b. you are divorced and not remarried, a widow or widower and not remarried, or you are married but your spouse is disabled and unable to care for themself.
(I don't understand the reason for these conditions, but that's what it says. If you are single and never married, or if you are happily married and both parents work, your parent's wages are not taxed; but if you get divorced and your parent cares for your kid, they are taxed. It's just weird.)
Since (most of the time) a parent's household employee wages are not subject to HHE tax, you do not issue a W-2. Your parent does not get credit in the social security system. Your parent reports the income on line 1b under "household employee wages not reported on a W-2."
