If you get there early enough, you can park ($25) on 1st Ave next to the pedestrian bridge. There are three lots there, and it puts you close to the stadium and all the restaurants in downtown. If you want to tailgate, park on the stadium side of the river, drive down South 1st Ave. and the parking lots line the right side of the road ($20).
ETA: If you are tailgating and drinking beer, the first lots on your right off South First have porta-johns, lots further down you can use the woods next to the train tracks.