The further you're away from Broad Street, the more you'll need a car to commute. If you live within walking distance of Broad Street, you can take the subway to Allegheny which is the subway stop right outside the dental school.
There are 2 available parking areas near the school, one near the school, anther a little further down closer to the hospital. There is also street parking (free) that you can do. A lot of students do it, but a lot of students don't (for obvious reasons). Risk/reward.
Edit: because you have a car, I don't recommend living in the city (cuz you can't use your car lol). Spring Garden, Fairmount, Fishtown, Northern Liberties, Roxborough, Manayunk, East Falls, etc. Maybe South Philly but parking will be tough for you car.