State parks are legal.
More info here.
Little fuzzy still, but local park's should be exempt through state law.
No fuzziness on local parks - it's legal. Municipalities cannot ban guns.
§ 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.