Fairfax County, Virginia
For many parks, yes. A complete list of parks that allow letterboxing and rules they must follow can be found at the website for the Geocaching, Fairfax County Park Authority.
Virginia State Parks
Guidelines for letterboxing in Virginia State Parks can be found online at Geocaching and GPS activities, which also includes a link to applications to plant a box. A permit is necessary if you want to plant a letterbox.
Petersburg National Battlefield
Letterboxing is not an authorized activity on the park grounds and boxes will be removed.