First off, don't say they're the same thing! I don't care about "common usage" they are NOT the same!
Umbrella, used to block rain. Made from a single water proof, air tight material. Usually made of rugged materials including metals and nylon. Yes, they block the sun as well. And they also block the wind making them a bit of a hassle to use if it's windy.
Parasol, used to block sun. Made from a wide variety of materials ranging from cotton and lace to rice paper and bamboo. These generally don't work well at blocking the rain. They block the sun fine. And they allow for air to get through making them lighter and easier to use when windy. They're also usually much more decorative and hence often times more fragile than a rugged metal and nylon umbrella.
More importantly, people look stupid walking down the street on a clear, hot, 90 degree, sunny day with a dang umbrella over their head!
To quote U.S.S. Enterprise Chief Engineer Montgomery "Scotty" Scott (the late James Montgomery "Jimmy" Doohan) in Star Trek: The Motion Picture (1979), "How many times do I have to tell ya? The right tool for the right job!"
We seriously need more Goth in South Florida to educate these simpletons in such things as parasols, hoods, and "outerwear" because they're doing it all wrong!