If you're interested merely in parking while visiting (rather than camping), have you tried giving the museum a call? They may have suitable parking available, or be able to suggest what would be appropriate. School groups and others with busses do go quite regularly to the museum; it's not completely devoid of infrastructure for large vehicles.
If camping overnight, I would probably tend to use the St Louis RV Park and walk the few blocks. If camping for a spell, I'd probably camp elsewhere and figure out public transit or whatever to visit the museum.
And in my opinion, the glowing reviews are fully deserved; it's a unique and wonderful place for children and adults willing to see things as children once in awhile. I thoroughly enjoyed my visit a couple years ago (by car, so parking was simple) and would love to go back sometime.