Parking lots *on* government facilities are generally, if not always, free. In fact, I don't even know if the agencies that own the parking lots/garages have the authority to charge for parking if it was warranted.
Note that this does not include private buildings in and around DC which are partially or fully-leased by federal agencies - a not uncommon arrangement.
Your policy recommendation (close the lots and institute a parking fee) wouldn't make it past a first-level review at the Pentagon, let alone Congressional oversight.
The only way I could see surface lots closing is for a replacement with parking garages (i.e. retain or grow the amount of parking spaces, but with less square footage involved).
Because if you close the lots - then what do you do with them? There's already a full transit hub there, so there's no real growth opportunity for mass transit. In theory you could develop the land by selling it off, but I can't see that happening for reasons related to security and the real estate market.
If you're going to get rid of *any* Pentagon-owned lots, it would be the ones on the south side of I-395, along Army-Navy Drive:
1) They're the furthest away and hence least convenient
2) They don't hold too many spots (although they do have parking for motor coaches visiting the 9/11 memorial, and also the Pentagon Reservation shuttle bus fleet)
3) They're the most likely to be possibly developed in the future, along with the Amazon City/National Landing/Bezosville developments.