I think that your parents are probably best off if they let their agent know that you are driving the car. In my case, my 13 year old Accord is still insured and titled to my parents, but I am listed as a driver on the insurance so I am covered. However, since I am not technically listed as the "primary" driver, my rates are lower (~$800/yr full coverage, $200 deductable). As for insurance following the car, the way I understood it (in OH) was that if I had to borrow a friend's car and something were to happen, my insurance would pay for the damages, and my friend's insurance would not have to pay for anything. In the event of you getting into an accident with your parents' car, they are much less likely to get "reamed" by the insurance company if the insurance company knows that you are driving beforehand. I'd say to notify the insurance company, an ounce of prevention is worth a pound of cure.
Zenmervolt