I don't know of any dedicated all terrain parks in VA though they may exist. Closest one I've heard of is Chaos in WV:
http://chaosoffroadpark.com
I've found some fairly rough trails in the past out scouting hunting areas in the GWNF, that took me through mud, water, rocks that sort of thing. Its been awhile and I wasn't looking for rough trails, I was looking for likely places to hunt, so I couldn't really tell you exactly where, but those trips put my vehicle (at the time it was a Pathy) through its paces.
Similarly, a few years ago hunting in late small game season I was up in the Rapidan WMA, where the main road that winds in and out of the SNP and the WMA is reliably rough. I hadn't shot anything, and only had a few hours left of daylight, and rather than head home I decided like an idiot to take the vehicle onto a sort of logging trail that circled the perimeter. There was thick ice over deep muddy ruts. I drove for a couple of miles on that thing and it got worse and worse, and then finally it sort of disappeared on a wicked slope. Big timber, getting dark, and I figured I was now officially screwed. I did finally got it turned around and out of there in low gear, but I won't try something like that again without another vehicle or at least a buddy to get out and help pick a line or dig or whatever.
There's always a tricky driving problem out there if you go looking for it.
. Just follow the law. No off-roading on national park land for example.