My mother is (was?) a member of the Smithsonian Institute, and she has mentioned to me that there was a time that as a Smithsonian member, she was able to drive and park for free in the garages at least under and around some of museums on the Mall (if not all of them). That would have been the 1980s and early 1990s. My guess is that the privilege went away post WTC (93), Oklahoma City, and the African embassy truck bombings. My understanding is that those events were all big for changes security and design around federal buildings in the DC area, and that 9/11 only accelerated it.
Slightly related, DDOT's Circulator bus now has a route effectively circling the Mall, with an origin/ending at Union Station. And at $1 (and free SmartTrip transfers) much cheaper than the old Tourmobile, which had for a long time had a right of first refusal against mass transit on the mall. The old Tourmobile buses can still be seen in the area, operating under the Martz brand.