Parking at uni of ExeterWatch
Generally I wouldn't recommend it, plus it's not really necessary or useful since most of what you need is available in a fairly compact area around the campus and city centre (which are very close). Unless you foresee needing to go to the Vale Shopping Centre or Sowton Industrial Estate regularly (or live in those areas maybe), I can't really see it being worthwhile even if you can get a permit for on-campus parking, unless you have some condition that requires you drive to the campus or have caring responsibilities meaning you have to have access to a vehicle while on campus in case you need to go at short notice. You probably wouldn't be granted an on-campus parking permit otherwise anyway, and parking in the city is going to be difficult and/or expensive otherwise.
I have/had my car during second and third year, and it is helpful for shopping and day trips but it's not really needed. The large majority of students don't have a car, and do just fine. If having a car is a big deal, make sure when looking for houses you get one with either a drive or guaranteed permit.