I've never fast passed a parade and done ok.
There is an outdoor seating area in Pecos Bill Tall Tale Inn and Cafe that is slightly fenced off from the street (the street is higher). It's usually empty and really easy for younger kids to see over the "fence'. It's right near the start of the parade so you'll want to be there before it happens and it's a quick service place so a nice spot to grab something cold to drink and take a rest (it is a outdoor eating area). It's also good if you happen to be walking up from where the pirates of the Caribbean is located because there is a side entrance to get to it when the street is already blocked off for the parade. It's the resturant where there is no sidewalk and most stop before getting to it since you can't walk the street near parade start.
Otherwise, I have great luck all the way by the train station at the entrance. parade takes about 20-30 minutes to get around to that spot. This is more often the spot I see the parade as I tend to just be getting to the park as the parade starts so we just wait there, enjoy and start our afternoon when the rope droppers leave for us to have more room. The spot next to the "macy's style windows" tends to remain empty but there is some tours that used to end when the parade started so they would take that spot. now that the parade is at 3, I think it's usually pretty empty.