The Blue Ridge Parkway is sister to Skyline Drive, which some (not us) would say is the raison d'jtre for Shenadoah National Park.

If it's bucolic country backroad you're after, the Shenandoah Valley is prime for aimless exploration. U.S. 81 is the interstate that runs through the area, but, well, it's the interstate. State Highways 623 and 613 run parallel, and are much more the lazy way to go. Up the ridge, State Highway 678 runs roughly parallel and in between the South and North Forks of the Shenandoah River. This road is in the forest proper, and offers good views and a chance to unload the family for a tour through the Luray Caverns, which are touristy but satisfying.

Love 'em/hate 'em, forest roads can offer splendour, particularly for 4-wheel drives. Check at ranger stations for some recommendations, as well as the must-to-avoid ways.