Quibble: the resort pools provide beach towels free of charge; the water parks only RENT them, for $2 towel (hotel towel size).
Lots of folks borrow towels from the resort pool areas and take them to the water parks; it's frowned on, but people do it.
We bring our own towels for water parks, but not for the resort pool; we just use the hotel towels for that. (2 reasons for this: unique towels work better to mark your spot at a water park, and not using them at the hotel means that they have a chance to dry before we head home -- wet towels on a plane can get pricey.)
I also bring a couple of swimmer's chamois to use in the theme parks after wet rides.
The gift shops at the water parks sell beach towels, but beware of buying them there -- thieves like to target brand-new Disney-themed towels left out on chairs.