Natural Bridge has been in private ownership since Thomas Jefferson purchased it in colonial times. It is a private tourist attraction, not part of a state or Federal park.
Many historic parks are also natural sites, such as Leesylvania and Sky Meadows state parks. Except for the Battle of Fredericksburg in 1862 and the siege of Petersburg in 1864-5, most Civil War battles in Virginia were fought on farmland away from the cities. Today, those battlefields are managed for their historical aspects, but are becoming valued as large blocks of open "green space"contrasting with creeping suburbanization.
The National Park Service classifies its parks as natural, cultural, or recreational areas. Activities that are appropriate for one type of park are not always compatible with others.
The national wildlife refuges in Virginia face the same conflicts, and are now completing "compatibility" studies. The U. S. Fish and Wildlife Service took over the former Harry Diamond Laboratory at Occoquan in the mid- 1990's. It soon chose to remove the buildings and towers constructed by the military, rather than convert them into visitor centers and nesting/viewing platforms. Tight restrictions were also imposed on scientists and educational groups visiting the new refuge.
This caught local environmentalists by surprise. They had blocked the county's plans to convert the base into warehouses or an employment center, facilitating the creation of a refuge instead. They expected access to various sites would be as easy as when the military was still in control, rather than more-restrictive. However, the new refuge administrators determined that wildlife conservation objectives required close monitoring of school groups and scientific collection, even if it cost the refuge some support in the local community.
No national wildlife refuge in Virginia permits camping. Instead, visitors are encouraged to hike, watch birds, study nature, and engage in other "passive" recreation activities. At most refuges, hunting and fishing are also encouraged, since many refuge acquisitions were financed by hunters purchasing "duck stamps."
The state Department of Conservation and Natural Resources manages 34 state parks and 40 natural areas, totalling about 1,000 acres.1 The Virginia Natural Heritage Program inventories natural areas throughout the state, maintains the registry of natural areas, and manages a Natural Area Preserve System.
Natural areas are left undeveloped in most places, while state parks offer cabins and campgrounds and picnic sites. The parks were developed to provide recreation opportunities, especially swimming. They are great places for family reunions, where the focus is on meeting and greeting other people rather than on the natural setting of the park. There are limits to the type of recreation permitted at state parks, however. In the summer of 2000, a proposal to create a string of new golf courses at state parks created a backlash. The reaction forced a reconsideration of what activities are compatible at those state facilities.
Local parks, of course, are often "active" recreation sites. They offer baseball fields, soccer fields, etc., sometimes developed in synch with local school systems or Little League groups. In addition, small towns almost always have a public gathering area, equivalent to the "green" in New England communities.

Bottom line: Where you are helps to determine what behavior is appropriate. Don't cut trees for firewood or let your dog run loose at a county park in an urban area - but dove hunters in a state wildlife management area may use dogs to flush coveys and sit around a campfire afterwards to relax...