When we were there last January, there were no designated dispersed camping-boondocking campsites in the national park that I saw. But there is backcountry camping and trailheads with parking areas. You can get permits for those areas and might be able to get one where you can stay in your MH.
One place we saw that you could probably use is the gravel Maverick Road inside the NP. If you go a few miles south of the northwest park entrance (nearest Terlingua), the road is still wide and well-maintained and there are some side roads and wide spots. Farther south, Maverick Road gets rough in spots and I personally wouldn't take my fifth wheel, or your MH, on that stretch. Maverick Road goes all the way to the Santa Elena Canyon Overlook, which is where we got on it after taking the paved Ross Maxwell Scenic Drive down to the river.
We stayed at Big Bend Resort in Terlingua--big campground with gravel sites and a few trees, but it had hookups, cafe, gas station, motel, and WiFi in the "library".
"Retirement is the best job I ever had!"
2015 RAM 2500 4x4 crewcab 6.7L CTD; 2016 Rockwood Signature UltraLite 5th wheel