Someone should start a fad of carrying umbrellas on sunny days. After all, "parasol" means [protect] "against the sun".
I often do this. The important thing is to make sure your umbrella is a lighter color (not black or navy) because the dark umbrellas absorb heat. I use a light tan one. In the summer, you have to be prepared for rain anyway, so I always have the umbrella with me.
Take advantage of any opportunity to walk indoors. Walk through the shops instead of up Main Street. Cut through the shops in Frontierland, or Tomorrowland.
Have some hand wash towelettes with you and when you stop in the restroom, wash your face, etc., so you'll feel refreshed. Running the cold water into your hands or across your wrists does help.
Always take a mid day break back at the resort. Take a cool shower and a nap, and you'll be refreshed and ready to go for the evening. I always have to do laundry when I travel in the summer, because I really need that shower and change mid day, but it's worth it. If you are staying off property and don't want to take a mid day break, book a TS lunch during the hottest part of the day and take your time dining.
Always, always, always, drink lots of water. Even those of us who live here and are used to the climate have to drink tons of water. If you don't, you'll start to feel sick in no time. Ask for a cup of ice water any time you go to a CS restaurant, even if you are ordering something else to drink with your meal. You can use it to refill a water bottle, and the ice will help it stay cool longer.