From this Gallitan national forest web site on Parkside, I wouldn't have any problem thinking a DP couldn't stay there. The photo of the park shows the road runs parallel to the highway for a very short distance. Having stayed in many N.F. campgrounds with our 40' DP motorhome, this would be a 'go' for us. It does appear that it could be a very busy campground though so try to get there in the morning when folks are leaving.
Custer Gallatin National Forest - Parkside Campground
Here's another good web site that gives you a picture of each campsite. It looks like the sites are very well done and if you look at the entrance sign picture, the road is as flat as can be.
The other campground, Greenough, is practically next door so I'm sure it's just fine. Try searching for pictures on it, too.
We love national forest campgrounds.