Can a parent be liabile for an auto accident caused by their child who is 18 yrs old and driving the child's own car?

Not if those were the only facts (although the parent's insurer could be liable if they are covered by the same policy), but you could come up with a scenario where the parent could be liable.  If the parent served the child alcohol, for example, and the child got into an accident because he was inebriated.  If, knowing the child was incapable of driving safely for whatever reason (they were high, they had been having seizures, etc.) the parent did something to facilitate the child driving while impaired (for example, jumping a dead battery knowing that the child was going to drive while impaired, or maybe just giving the child keys which the child had lost).  If the child was not a competent driver but the parent helped him or her cheat to get their license, or bribed the person administering the driver's test.  If the parent had done something to make the car unsafe, like installing improper brakes.  You could probably thing of numerous far out scenarios where the parent would be liable, but any liability would depend on the parent having done something beyond just being the parent of an 18 year old bad driver.