In my opinion "YES " you will miss the best part of the parkway, 6 days is long enough to see the north part and Shenandoah N.P.
Shanendoah N.P. is a really nice relaxing park as well, we have spent 3 days on three different trips in Shanendoah, it has the most black bears per square mile of any park in the lower 48 states, so practice bear food storage saftey.
with that said the entire parkway is really nice, we love the BRP and travel on it as often as we can, we spent 18 days on the parkway in 2011 and could have spent more time.
the southern end for sure south of Ashville is the highest elevation and thus my favorite part.
Just think of it this way, you have a good reason to go back for another visit.