there's an easy solution to that. Don't watch the first parade. skip it, go do some rides in areas away from the parade or see if they have a show you can watch during it. Then watch the second parade when the crowds have thinned out tremendously and you won't have to fight for a spot- even on the square.
Parade time isn't a bad time to check out the train, of if you have people that want other rides, it's a great time for Fireman's Landing, Expo, Barn Swing, or Outlaw Run. :-)
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We went on a Friday night the weekend before the Thanksgiving holiday weekend and we waited until the 2nd parade. It seemed just as crazy and packed as I have seen it up on the square.