If an 18 year-old child causes the passenger in his car to be hurt in an accident, would their parents be liable?

The parents would not be liable based on the parent-child relationship, since the parents' legal right to control the child's behavior no longer exists.  However, if the car is owned by the parents, they could be liable on that basis, as the law considers anyone who is driving an automobile owned by someone else (whether related or not), with the permission of the owner, as the agent of the owner; this makes the owner liable along with the driver, for the driver's negligence.

When someone is held legally responsible for harm caused by someone else, it's called "vicarious liability."  There is always some reason at the heart of any sort of vicarious liability, some at least theoretical way in which the person who has to answer for someone else's mistake had some control over the situation.