For this, I would watch on the Pecos Bill side of the street in Frontierland. As soon as the parade has passed, simply turn and scoot back down towards AdventureLand. If the sidewalks are crowded right on the route as they clean up, you can also enter and cut through Pecos Bill / Tortuga Tavern and come out in Adventureland. Then from the entrance to Adventureland, head down the sidewalk by Crystal Palace and Casey's. You can get down Main Street either by hugging the wall side of the sidewalk or by simply walking through the interiors of the shops and get out of the park that way, well before the parade gets to the Hub.
(From the catalog of been there, done that.)