I know that at Disneyland they HAVE to run the parade in alternating directions when they run it twice. Unlike Magic Kingdom, they do not have a way to get the floats from the storage barn to the front of the park without going onstage. So when they run the parade twice, the parade runs from Small World to Main Street, the parade hangs out backstage for a bit, and then runs from Main Street to Small World. (I assume that when a parade is only run once per night, they relocate the floats after park closing)
This caused a problem during the ill-fated Light Magic. One of the ways they tried to speed up the show was to start the floats from Main Street. But because the charge cycle for the floats was longer than the overnight hours, they had to re-locate the floats mid-day using CMs to clear Guests out of the way along the parade route. (I heard that they had to come up with a recorded announcement to explain the floats and to advertise the parade/street show later on that night.) I'm sure those floats that were designed only to be seen at night weren't all that great-looking in daylight, either.
-Rob