Another vote for Pine Mtn RV Park. You might just be able to do without renting a car or UBER. It's 2 blocks off the parkway so you don't get much noise, but it is right in the thick of things.
As someone stated, that is the busiest time of the year, so you may have to go with wherever you can find a spot.
Someone suggested Anchor Down and it is great, but has a price tag to match it. Just across the late is Douglas Dam Headwater Campground, which is very nice, but has water/electric only. It's $25/night. We also like Smokey Bear in Cosby, but it takes 30 minutes to get into Gatlinburg and another 15 to get into Pigeon Forge, as well as Whispering River Resort in Walland, TN, also another 30 minutes out.