Most universities only have limited parking for students, and even then you usually have to buy a parking permit which will be based on a needs assessment. Finding a university with parking in/around London is even more of a stretch.
The University of Aberdeen (miles away from London, but meets your other criteria) offers Medicine, has student parking permits available and also has plenty of free on-street parking around the campus. The space I park in is about a 3 minute walk from the main part of campus, it's actually closer to some buildings than the student car park. If you are willing to live away from home, it might be a good option for you.