I'm not familiar with that particular park, but if they have any signs posted, or any "rules" prohibiting the carrying of firearms, whether carried concealed or openly, would be in violation of the state preemption law.
§ 15.2-915. Control of firearms; applicability to authorities and local governmental agencies.
A. No locality shall adopt or enforce any ordinance, resolution or motion, as permitted by § 15.2-1425, and no agent of such locality shall take any administrative action, governing the purchase, possession, transfer, ownership, carrying, storage or transporting of firearms, ammunition, or components or combination thereof other than those expressly authorized by statute. For purposes of this section, a statute that does not refer to firearms, ammunition, or components or combination thereof, shall not be construed to provide express authorization.