Yes, you can make it. I'm pretty sure the parade starts next to the Fire Station/City Hall so I would watch it from a spot near the beginning - either around Town Square or on Main Street. I don't know how long the parade is, but probably not much more than 20 minutes. Once the parade ends you can head out and walk over to the Contemporary. It'll take about 15 minutes.
Or you could take the resort monorail, but you'll need to allow extra time if you have to wait for it. The CR will be the first stop after the MK.