note - if using a hammock near a hut (what Shen folks call a shelter) it may be worth while trying to get the hammock hung so that it at least looks like it is on one of the official "tent pad" campsites - I did once see a Ranger ask a hammocker that had hung near Pine Field Hut to move because he was not near one of the official tent pads. That said the likelyhood of being caught is fairly low unless like Pine Field there is easy access to the Skyline Drive so that the Rangers can walk in easily.
I live about 20 minutes from Rockfish Gap and know the guy who manages the gear shop at the bottom of the mountain. I could ask if you could park there... Up at the top of the mountain there are plenty of empty parking lots due to a hotel and gas station that have closed down. I don't know if there are people who monitor, and tow though. How long would the vehicles be parked??? PM me for more details... It's gonna be sweet up there no doubt. I did my first hang up there this past weekend, it was cold but beautiful!