Sunscreen for your left arm as you drive. When I did a parade years back in Wichita Falls in my fordoor, my left arm got a severe sunburn on the top of it where it hung out the door window as I drove.
All the chronic stop and starting as you drove behind the band or the cheer leaders as they stopped to do their act of baton twirling or such was tiring with a long parade, but much better than being behind the horses and smelling what they left on the road in front of you for the duration of the parade. Get your place behind an interesting group and not the horses or, Lord help you, the elephants.
If you have the room, give the folks from the AARP or any vet group a ride. They can probably remember what riding in a Model A Ford was like in real life, so it brings back lots of memories for them. Kids are another option, but someone mentioned tossing out candy, which can be fun, but be sure it is allowed. In Vernon, TX, at the Summer's Last Blast, they will stop you when you do that as they feel it is a safety hazard with kids and adults rushing to the street and picking it up. Check to make sure it is okay, first.
Make sure your engine is tuned up and radiator working well before heading out.
Have fun and take lots of pictures or a good video. Bring a dog to ride and folk always love that. Make sure your horn works well, as the Ahooga sound is a must for a parade.