Federal law defines "facility" as "building."
I have the same problem when I go to the VA hospital. I know I can't carry within the building, but I used to be able to have firearms in the car and be able to proceed from home to the hospital, and then from there directly to the shooting range. Several months after Obama was elected, signs appeared at all entrances to the property saying that no firearms are allowed anywhere on the property. Interestingly, the signs cite 18 USC 930, which clearly defines Federal facility as follows:
The term “Federal facility” means a building or part thereof owned or leased by the Federal Government, where Federal employees are regularly present for the purpose of performing their official duties.
Since an outdoor, surface parking lot is clearly not "a building or part thereof," it does not appear that 18 USC 930 applies to other than buildings. However, I can't afford to be a test case.