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While I can't find the Department of Conservation & Recreation regulation on it (or any of their regulations, for that matter), they do appear to prohibit carrying of loaded firearms in state parks (except in hunting areas) by anyone who doesn't have a "valid concealed weapons permit." I find it irksome that DC&R does this by regulation, but I'm not going to argue that point.
It would appear to me, though, that carrying concealed in a state park is expressly allowed by anyone who can carry concealed elsewhere in the Commonwealth. Academically, I would argue that the DC&R regulation also allows open carry of loaded firearms by CHP (or reciprocal) holders, but practically speaking, I wouldn't be surprised to get hassled by law enforcement for open carry in a state park.
I apologize for my incorrect initial response, James, but it's apparent that you can certainly carry a concealed handgun in a state park in Virginia with your West Virginia permit.