New River Junction @ McCoy Falls (Big Falls Park) is a family campground. The same people who do the inner tube float trips.
As far as I know New River Park farther down the road is a private campground where you must own or rent a permanent site. They have a website if you search for it.
I do a good bit of wading around Parrott and from McCoy to the Eggleston bridge.
This past weekend the water was high, fast, and murky. I reckon Claytor Lake is still discharging a good bit of water.