here's an online map from Franconia Notch State Park site
http://www.franconianotchstatepark.com/franconiahikingfinal.pdf
there's a paved bike path labelled "Recreational Trail" on this map, marked as a dotted line - this bike path runs from Skookumchuck Trailhead to the Flume visitor lot
last saturday, i chained my bike to a tree at Lafayette Campground, then drove to the Whitehouse lot, which is about 0.1 miles north of the Flume visitor lot off route 3
from there, i hiked Whitehouse Trail 0.8 miles north to where it rejoins the bike path, then followed the bike path north to Liberty Springs Trail, up Flume Slide Trail, over Flume > Liberty > Little Haystack > Lincoln > Lafayette, down Old Bridle Path, then biked back to my car (3.5 miles on paved bike path)
parking in Whitehouse lot allows easier access to Whitehouse Trail, which is more enjoyable than hiking a paved bike path (ymmv) ... parking in Flume visitor lot will only add 0.2 miles, so if you'd rather walk the paved sidewalk, park in the Flume lot - knock yourself up
there's a parking lot near the Basin on the southbound side of route 93, but it's about 0.8 miles hike south (on the paved bike path) to Liberty Springs Trailhead, so won't save much time
http://www.franconianotchstatepark.com/directions.html
in hindsight, i could have saved biking mileage by parking at the lot near the Basin, but it was
fun! cruising down that bike path after hiking the ridge on a hot sunny day
Flume Slide was slickery in some areas, but overall pretty easy ... hope this helps !