State parks in Virginia are good to go for OC. So, by the way, are Virginia State Forests.
If by federal parks you mean National Park Service parks, they are also good to go. National Forests, places under the domain of the Corp of Engineers (like Lake Kerr), have different rules. There is a stiucky at the top of the Virginia page that covers your question in more detail than you probably want. But go read it anyhow.
Why are bothering the Governor? He has minions that deal with stuff like State Parks. Unless the Montgomery County Sheriff's Office has original jurisdiction in a state park within that county they are not likely to know anything about the rules. Go bother the minions who are responsible for dealing with state parks., and maybe look for the web site about Virginia State Parks, where they provide information about firearms.
As for National Parks, they have their own web site. The rules about firearms are right there.