Yes, parents with driving-age teens can not only add their child to their insurance policy but also their child's car. Many states require a legal adult to be included on a vehicle's title, registration, and any potential loans. In fact, adding a child to a parent's policy is generally a better idea than giving the teen their own individual policy. Parents can benefit from cost-saving discounts such as multi-car, safe driving record, and multi-policy. Teenage drivers will have an expensive premium more so without the benefit of their parent's existing policy.
In the event that the teen has many tickets or car accidents on their driving record, it will be very expensive and possibly shed a negative light on the existing policy. After many violations and accidents in a short period of time the insurance company will likely threaten to drop the policy if the teen isn't removed from coverage. In these cases giving the teenager their own policy may be advantageous.