It's only a five minute walk (ten at most) from the Rusty Parrot to the town square.

From personal experience, I prefer not to have to walk anywhere at night in Jackson during the winter. It can get very cold. Parking, however, is not such a problem in the winter, so as long as you have a car, you can drive the short distance from the Rusty Parrot to the center of town (it will take you about two minutes).