If you have access to a freezer before your trip to the park, I suggest freezing the water so it stays nice and cold as it melts throughout the day. We also do this with Gatorade bottles - so refreshing on hot days.
Normally, we'd do that as well, but the water at Disney is disgusting! Even some of the "filtered" water there is a bit off to me. So at Disney we bring a case of water in our car and put it in the mini-fridge in our room. Never had a problem bringing water into the parks.
We use the LifeStraw Go filtered water bottles. You can fill it up at any water fountain and it filters out the yucky stuff as you drink it. The bottle comes with a short strap and carabiner that makes it super easy to clip on a backpack or whatever. We have the 22 oz size and it's perfect! We got them from Amazon. So much easier than dealing with water bottles and much better for the planet!