Your lunch plans make what I'm about to suggest tricky, but...
On one occasion a several years ago (not FoF, but whatever parade they were doing back then), we saw the parade as it passed by in the breezeway between Adventureland and Frontierland (the one that's near Sunshine Tree Terrace). Once it had passed by, we moved quickly through Adventureland toward the entrance and caught it again near the train station. Of course, the park wasn't very crowded that day and there was no hub construction, so it was pretty easy to get from Adventureland to the front of the park before the parade came through. Just something you might try if you're daring enough.