Just another thought from the other side of it... since I've got teenage kids.

If you are stupid enough to hurt someone with your car, and get sued for more than your insurance is worth, your parents could lose their home & lots more. They may want you on a separate policy to protect themselves from that. OTOH, some states laws are set up so that doesn't protect them anyway.

Like I said, each state has different laws & each insurance company has different ways of doing business.