If you do the babysitting in their home and follow their rules, you are considered their household employee and they should be handling the payroll taxes, filing Schedule H and issuing you a W-2 at the end of the year.
However, any care you provide in your own home is self-employment income to you and filed on Schedule C. Since they did not do what they should have done, you can choose to file all the income on your own Schedule C and pay the self-employment taxes and have them reimburse you for their portion. Technically, they should pay half and you pay half since you are an employee. You can decide with the family the best way to handle the 2020 return.
If you want them to issue you a W-2 next year and handle the taxes for 2021, they need to get that set-up now and start withholding. The IRS just wants the income reported and the associated taxes paid.
You can be both their employee and self-employed if you provide care in your home and in the parents' homes. But yes, you are an employee for the care you provide in their home. You could insist that they issue you a W-2 for 2020 and the IRS would agree, but that doesn't really help you or them at this stage. If you provide care in your home for other children, that income goes on Schedule C and you can report the income you earned while working in their home on that same Schedule C for 2020. And yes, if you want them to give you a W-2 next year since you are their employee, they should start paying you as an employee which means paying and/or withholding payroll taxes on your behalf. @whitneysanders95
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