Can you take the Blue Ridge Parkway from Shenandoah National Park to the Great Smokey Mountains National Park?

No. From The Skyline Drive (SNP) you would need to travel west minimum of 2 hours through the Cumberland Gap .Having driven the Blue Ridge Parkway (using Roanoke as my gateway) I can tell you the drive is very pretty and the roads are well-mantained.If you take the Parkway to its Southern Terminus you would need to drive through the Mountains if you exit in VA the Altitude in much lower its your choice.