Why? The teenager is more than likely going to be going for the coolness factor and the parents are going to take a more practical approach in terms of insurance cost, the safety record, the reliability factor and overall yearly operating cost.
If the vehicle for the teenager is brand new the cost of insuring it might be higher because if you need to replace or repair the vehicle it is probably going to be more expensive than an older used car, it is important to do the research because it is a known fact that some older cars have higher insurance rates because the vehicles are stolen more often than other vehicles for their parts.
Before discarding the idea of a new car completely it is wise to research the safety features and history of the new vehicle because they might override the cost by getting discounts from an insurer.
The bottom line to the perfect car for a teenager is to do your homework (and you thought they would be the only ones doing the homework) and research the different models for safety and reliability and contact your insurance agent to find out which models offer the best yearly rates in terms of insurance cost.