I love this site for this kind of question:

http://washingtondc.bestparking.com/index.php

I've seen guides online for this. e.g.,
http://dc.about.com/od/transportation/a/ParkingNearMall.htm

Metro is not an option, I know it's the Cherry Blossom festival, and I'm willing to pay for expensive parking fees if necessary. Where around the Smithsonian will I be able to find parking this Saturday? Thanks!