Employee or contractor is not a choice the parties make. Publication 926 states the test directly: "The worker is your employee if you can control not only what work is done, but how it is done." Full time or part time does not matter, nor does hiring through an agency, nor whether pay is hourly, weekly or by the job. The mirror image is equally clear. A worker who controls how the work is done, provides their own tools and offers services to the public is self-employed. A childcare provider who works in their own home is generally not your employee. And where an agency both provides the worker and controls the work, the worker is the agency's employee, not yours.
Not everything done in a home is household work. The publication draws the line at the nature of the service rather than the location: work performed as a private secretary, tutor or librarian is not household work even when it happens in your house.
Several relationships are excluded outright. Under section 3121(b)(3), wages you pay your spouse, your own child under 21, or your parent are generally outside Social Security and Medicare tax, with a narrow exception that can pull a parent back in where the employer is a widowed or divorced parent with a young child at home. Section 3121(b)(21) separately excludes domestic service by anyone under 18 whose principal occupation is not household work, which is what takes the neighborhood teenager who babysits on Saturdays out of the system. Wages excluded from these payroll taxes are still the worker's taxable income.
The two wage tests are different tests, and missing that is the commonest error. Social Security and Medicare tax attach once you pay a single household employee cash wages of at least $3,000 in the year. Section 3121(x) reindexes that figure annually against the national average wage index and rounds it to the next lowest multiple of $100, and it is tested per employee rather than across the household. Federal unemployment tax runs on an entirely separate quarterly test: section 3306(a)(3) makes you an employer if you paid $1,000 or more in cash wages for household service in any calendar quarter of the current or preceding year, counted across all your household employees rather than one at a time. A family can therefore owe one of these taxes and not the other.
The worker's side of the ledger is the part paying cash quietly destroys. Reported wages build the Social Security and Medicare earnings record that future retirement, disability and survivor benefits are calculated from, and state unemployment coverage usually follows from the same reporting. A household worker paid off the books has none of it, and no pay stub to show a landlord or a lender.