If you really want to save on parking and don't mind walking, you can park on Mount Washington, take the Mon Incline down and walk across the Smithfield Street Bridge. This has the added attraction of avoiding whatever PennDOT is doing with the Fort Pitt Tunnel this week.

(You can also do the "park on the other side of the river for free and walk across a cool old bridge" thing in Cincinnati, but I wouldn't try it in St. Louis.)