While i was a medical student, a nursing student was stabbed on her way out of A&E to the bus stop late at night, i was working late that night too. That was central manchester, not bristol, but cycling through St Werburghs at 2 am on the way home from Frenchay ED was equally risky!
I drove a heap during my clinical years, but it made me feel much more secure with late finishes etc.
Healthcare students tend to have placements out and about, and long hours, so not necessarily meshed with public transport (bristol uni send them as far as yeovil and taunton fyi and ive done both of those by train after qualifying - on Mot day - and its not conducive to study!)
What about non-central universities? Would you have the centre of bristol only available to bristol uni students and exclude uwe? That would be likely to hit mature students and students with families more heavily. Are you happy with that? What about OU? Postgraduates? Would you have student parents be late after dealing with nursery drop offs?
Passes would be ridiculous too. There are many parking passes for campus unis already, and students pay up if they can get one or park on the public highway. Outside of class, they are adults who pay insurance and road tax, so what could it have to do with anyone else. They still have the same other journeys to make as anyone else.
Bristol parking is atrocious, and im not sure residents' permits are the way forward tbh, but I chose not to move to clifton/redland because of the parking. Even without the student cars, you still have commuters leaving cars on residential roads, and the permits will make it more difficult to make occasional visits to these areas of town, whether for shopping, meeting with friends or whatever. But it doesnt follow that its a student problem.