Anyone can be sued for just about any reason. That said, the owner of the car is typically considered to be the responsible party. In Texas and other states, the "owner" of a vehicle in auto accident lawsuits isn't merely defined by the name on the vehicle title. An owner can also be anyone who has deciding control over the vehicle's usage.
People who are driving or parked at the time of a hit and run accident are rarely held accountable unless they performed some sort of action that caused the accident. You fail to clarify whether your son was the victim or perpetrator of the hit and run. If your son committed the act, other parties will hold him liable for the event. In the case of non-ownership where you have control over the vehicle, you could also be held liable in what is called a negligent entrustment scenario. The party suing you must prove that your son behaved negligently, that you had primary control over the vehicle's usage and that you knew that a potential for negligent behavior existed at the time you allowed him to use the car. Your description doesn't make it seem like that's the case in this situation.
If your son was in an accident where his car was hit by someone else who left the scene, the other driver would be the one held responsible unless your son behaved negligently during the event. Of course, if another person's property was damaged in the accident, even without any negligence on your son's part, the property owner might attempt to file a claim against your insurer and sue everyone listed on the insurance policy if that attempt fails.
When you have these types of concerns, the best course of action for peace of mind is to speak with your auto insurance agent or an auto accident attorney.