Road biking parallel to the AT is an interesting idea. I have never heard of anyone doing it, but the concept sounds interesting.
Down south, much of the trail is roughly parallel to the Blue Ridge Parkway between the Great Smokies and Rockfish Gap. But, you will need to go off the parkway a fair bit to get into the great trail towns like Damascus and Hot Springs. Then, the AT crosses the Skyline Drive over 20 times in Shenandoah National Park.
In Georgia and North Carolina, the AT is in the middle of national forest lands, so there probably isn't a good road close.
Through Pennsylvania, the trail is on a ridge, but there is probably a good route through the valley, and you can hit the good trail towns like Duncannon (Doyle Hotel is legendary) and the Water Gap. Through New Jersey and New York, roads are never far away, so that shouldn't be problem to plot out. Starting in Connecticut, Route 9 roughly parallels the AT until Rutland Vermont. Then, follow Route 4 over to Hanover. From then on, it's going to be tough to parallel the AT closely.
Sounds like a great follow up for someone who has hiked the AT.