OP: Do as they ask, and do it early. Then ask if you can throw a few others in just to see what happens.
At some point you will have to decide whether to apply behind their back if they say no. Scholarships require early applications usually.
I’m not for splitting up families, but if your choices are Harvard or working at Starbucks, I might recommend striking a blow for independence. If they really won’t bend, you will need to fend for yourself and your own future.
Some high schools require safeties. Perhaps a little white lie won’t hurt?