State laws differ but in general, yes your son could be held responsible for the damages he caused to the vehicle he was driving. As the driver he was the one in control of the car when he hit the parked car. He may have been doing a favor but received permission to drive the car and was the one operating it at the time of the collision with the parked vehicle. 

If you want to know about specific liability laws for your state I would recommend getting legal counsel. You can get answers to your automobile insurance questions and get an insurance quote here with us.