Liability of letting teen drive pengers?

I got my license a couple months ago, and I'm 16 years old. We have 3 cars for 3 drivers, yet they are all registered under my dads/moms name. I get to drive my dads BMW, and he put me on the policy for the car. My mom is scared that there is liability issues driving with me driving other teens. She lets me drive with my best friend, because she has spoken to his mom ect. My mom claims that if I get into an accident with others teens, that they can sue you. I don't understand this though, doesn't the insurance pay for any penger injuries up its limits. Similarily, if I were to hit a pedestrian or get into an accident injurying the people in the other vehicle, couldn't they sue us? My mom claims since the car is registered in my dads name it poses a problem. This is in Florida, does anyone know if this is true?