To the best of my knowledge, you may only have a firearm in a Virginia State Park if you are in areas designated for hunting or if you have a valid concealed handgun permit.
Virginia State Parks
Managed and regulated by the Virginia Department of Conservation and Recreation
Virginia State Park Preliminary Summary:
Open Carry: OK in areas designated for hunting
Concealed Carry: OK with a valid permit issued pursuant to § 18.2-308
Unloaded and cased: OK in areas designated for hunting
References are available in the thread referenced above.