UPDATE 02-Jan-2007 The 2 story parking garage has greatly improved parking! It is located directly behind the Visitor Center.
Parking is an absolute nightmare in the city. It is best to get there real early or late in the afternoon to find a spot by the Visitor Center.
When I was just there in December 2005, they are building a two story parking deck. I am unsure when that will be completed.
You may get lucky and find a spot along one of the back side roads as well.
Caution- I wouldn't park at Ripley's because you could get towed. Back in the day, we used to park at Ripley's but they have since gotten wise to the fact