Since you were paid cash, sounds like they didn't take anything out, even social security. So your dad isnt correct, the filing limit for that kind of income is four hundred dollars for the year. You won't owe federal income tax, but will owe something called self-employment tax, for social security ane medicare. You'll owe around $700. That's not because you're a dependent, but because no ss and medicare were taken out. If they had been, you wouldn't have to file.
The $5700 limit to have to file is if they took out tax and gave you a W-2. And if you worked in their home, that's how you SHOULD have been paid, but that's another question.