My mom and step-dad decorated their truck in red, white, and blue and we were off to be in the Baxter Parade in Fort Mill, SC. It was a terrific and patriotic time enjoying the small town feel of Baxter and practicing our princess waves as we threw candy to the parade watchers. My daughter is an expert princess waver....lol!
I took my Nikon D5500 DSLR camera with the 18-55mm kit lens. I usually prefer my 35mm 1.8f Nikon prime lens (prime lenses do not zoom in or out) because of the ability to open the aperture larger and the better portrait quality. I also like prime lenses because it gets me up and moving as I take pictures. I feel more of a connection to my subjects when I do this. However, the prime lens was not necessary since I was not taking pictures of clients, but my own family. I also felt like the zoom was the better option when being a part of a parade. The ability to zoom in so that I wasn't running around everywhere trying to get the right placement for my pictures was important. Although the parade watchers might have found it entertaining as I tripped over myself!
My best suggestion to you, if you are taking pictures in a parade, is to make sure you have a high shutter speed (1/500 or more) so that you are capturing motion and don't get a lot of blurry images. If you need to raise the ISO (mine was set at 800, but I usually like to keep it at 200 for sunny outdoor images) in order to do this go right ahead! I would personally rather have a little more grain in my pictures than blur....unless you're being artistic and want some blur in your pictures! When it comes to aperture you will want to have it at f4.5 or higher number (smaller opening) so that you get more depth of field (things closer and farther away from your focal plane are in focus).
Note: These suggestions are not the only way to take pictures of parades. You should try different settings and see what works best for you and the feel you want to convey in your images. Have fun and get out of your comfort zone!