Code of Virginia § 46.2-1231, and specific signage requirements can be set by city/county ordinance, see § 46.2-1232.
The pertinent part of the code section:
The owner, operator, or lessee of any parking lot, parking area, or parking space in a parking lot or area or any part of a parking lot or area, or ofany other lot or building, including any county, city, or town, or authorized agent of the person having control of such premises may have any vehicle occupying the lot, area, space, or building without the permission of its owner, operator, lessee, or authorized agent of the one having the control of the premises, removed by towing or otherwise to a licensed garage for storage until called for by the owner or his agent if there are posted at all entrances to the parking lot or area signs clearly and conspicuously disclosing that such vehicle, if parked without permission, will be removed,towed, or immobilized. Code of Virginia § 46.2-1231 (2006).
Later in the code section, it authorizes boots as well.
I highlighted in blue the most pertinent part: permission. You must park according to their demands, or you do not have permission to park there. Following their rules becomes a condition precedent of their permission. Don't follow their rules, you don't have permission. Without permission, they may tow or immobilize you.