Why don't you PM Windchime or Webmaster Eric - they probably could find a place in the PJ Housing complex/shop area for you to park it. If that doesn't work there may be a spot that the Campground Host could give you in Chisos Basin or RGV... TWWG
I think you are assuming a bit much here. As soon as an employee lets someone do this, the gates are open on more requests and expectations of similar treatment. Can you see the problem? And that is without considering any liability issues.
I don't think it's going to happen in either the housing area or a park campground. Park employees bend over backwards a lot farther than anyone else will to assist tourists, but don't abuse this and put them on the spot on a personal level. Campground hosts aren't going to do it either as there's no provision for storing unattended equipment there or anywhere else in the park.
However, if you go to Stillwell's or Study Butte I have no doubt someone will let you drop a trailer for a reasonable or not-so-reasonable fee. Support the local economy.