Parking has been a problem not just at Seattle University but on First Hill and Capitol Hill for years and probably will continue to be even with the forthcoming expansion of public transit. If by any means you can get by without a car, do so, or you will have to be prepared to occasionally arrive 10 minutes later than normal and find the lots full. That might seem like an exaggeration, but there aren’t enough spots for the number of students who do drive, and given how expensive passes actually are, it is probably a better idea to just pick up one of the discounted bus passes from public safety and be prepared to stand at a Metro stop every morning and evening.
That being said, if you do choose to drive, the hill above campus is your best bet to find a spot, but even then you have to get there early, and you are never guaranteed a spot. Don’t even bother trying anywhere along 12th Avenue or Madison, and even Jefferson can get to be a headache on occasion. The Residential Parking Zone program is an option from the city of Seattle, but those passes are only available to residents of a neighborhood, and they cost about $45 for a single cycle (usually two years, though some last only one). There is also always the option of parking in a lot, but that can be more of a hassle than it is worth sometimes.
In the end, the park and ride system might be the best bet for those living off campus (especially far out in places like West Seattle or Rainier Valley), as you would still have the independence that owning a car can bring but would not have to deal with the nightmare that is First Hill/Capitol Hill parking. If you’re living on campus, don’t bring one unless you really see it as essential to your lifestyle. It is easy, however, to learn to live without one.