They just worded it poorly, Jim. The governing Virginia Administrative Code is
4VAC5-30-200, which states in part:
No person except employees, police officers, or officers of the department shall carry or possess firearms of any description, or airguns, within the park. This regulation shall not apply in areas designated for hunting by the Department of Conservation and Recreation. This regulation also shall not apply to the carrying of concealed handguns within state parks by holders of a valid concealed handgun permit issued pursuant to § 18.2-308 of the Code of Virginia.
Now, one could argue, as I often do, that the VAC isn't law, but that's a different issue entirely. The bottom line is, they're talking about CHP holders carrying concealed.
ETA: Try as I might, I can't find anywhere that states what the penalty is for violating this regulation, or any State Parks regulation, for that matter. If there's no penalty, then it's a matter of, "Okay, I broke the rule. So what?"
~ Boyd