If you are not an independent contractor, then you are a household employee. They need to declare your income on their tax return, pay household employment tax, withhold social security and medicare tax, and issue you a W-2.
Here's a recent answer I gave from the parent's point of view, be sure to read the comments as well as the main answer. https://ttlc.intuit.com/questions/4121796-hi-i-will-be-paying-a-nanny-to-watch-my-son-i-know-she-wil...
Now as a practical matter, how much will you put up with to keep your job? If they pay in cash and don't issue you any tax forms, you have two choices.
1. File as an independent contractor even though it is incorrect, and eat the extra taxes.
2. File with a form SS-8 which is a request to have the IRS investigate your tax situation and determine if you are an employee or contractor. You will only be responsible for the employee half of social security and medicare tax, but I would be surprised if you kept your job once the family got the letter announcing the investigation (which will end with them owing taxes and penalties.)
(There is a third choice which is you can commit tax fraud with them. If they don't issue a W-2 as a household employee, and if they don't issue a 1099-MISC form -- which private citizens are not required to use -- then there is no IRS notice of your income. You could potentially report less than the full amount of your income -- working "under the table." For obvious reasons I don't endorse this course of action.)
You can also file a complaint with your state labor board. Unless you live in the home as an au pair and they provide room and board, you are probably subject to ordinary wage and hour laws including the minimum wage and overtime for more than 40 hours worked per week. But naturally, filing a complaint may lose you the job.
