Try the near East side, like Willy st/Atwood area. It's the closest thing to a gayborhood, but it's also kind of the aging hippie neighborhood, and it has a lot of cool restaurants and cafes and such. And it's home to one of the city's few gay bars.
There's public parking in the Park St. ramp, but it gets expensive real fast. Street parking is similarly expensive and usually has a time limit. Spaces in lots are usually reserved for employees/are hard to come by. So, if you have the money, it's doable, sure, but biking and busing will save you a lot of money, especially with free bus passes for students.