The Parkway can be tame (TN 66/US 441) can be tame, depending on when you're traveling on it. I've witnessed Sunday mornings as a big no-no, but haven't been that way since they finished expanding to six total lanes during that timeframe to give you an accurate feel. The DDI does feel like it helps get traffic through a bit better, though.
Get to Gatlinburg as early in the day as you can. Traffic and parking get worse as the day progresses (like everywhere else), but it's bad on that main drag heading into the park. If you're coming into Gatlinburg from the park, it's possible to take a bypass around Gatlinburg, but you still might have fun once you get towards Pigeon Forge.
Veterans Blvd kind/sort of parallels the Parkway, and goes by Dollywood. It also ends on the east side of Sevierville, at 411.
Elizabethton has the distinction in which 321 goes South heading east or west from town:
http://www.millenniumhwy.net/interestingsigns/tn/US321-Tennessee/If you wanted to take 321 to Cosby/Gatlinburg via 81, you could also hit either TN 93 (Exit 50), TN 172 (Exit 36), or TN 70 (Exit 30) and pick up 321 in Greeneville. Each route is two-lane. Leaving Greeneville, you start off two-lane, but switch to four-lane a little bit outside of town.
The aforementioned 25E is another option, to Newport. Both of these feel slow to me, but depending on traffic on the Parkway, may be a bargain. You'll need to pick up 25W to go past 40 to get to 411, if you want that route.
You can also avoid some of the Parkway via Douglas Dam Rd to Deep Springs Rd (Exit 412) heading back on 40.
Have fun.