Preview from Parkside Cabin Rentals's listingHomes that state their distance to a park entrance, from professional managers in Gatlinburg, Estes Park, and Moab.
One rule for this list: the listing itself says how far to the entrance. Not a ranking. Three parks to start.

Every gateway town says it is near the park. The useful listing says one mile from the Fall River entrance, or 1.8 miles to Beaver Meadows, or across from the Greenbrier entrance. That sentence is the difference between being on the trail at eight and sitting in the entrance line at ten.
Ten homes, three parks, from professional managers with their own booking sites. Every card quotes the listing's own distance claim. Park entry rules change by season; the official page is linked under each park.
Preview from Parkside Cabin Rentals's listing
Preview from Parkside Cabin Rentals's listingReserve on Recreation.gov; two-hour entry windows.
Preview from Windcliff Vacation Homes's listingFiery Furnace and Devils Garden campground still need reservations.