Linda,
Yes, you are absolutely allowed to bring in your own bottled water into the parks! You are also allowed to bring a cooler into the parks, you are just not allowed to use loose or dry ice.
Of course, if you need to refill those bottles during the day, please know that Walt Disney World has water bottle filling stations in each park and are adding more periodically. You will find that there are several of these water stations in each park, and they seem to add them as they design or renovate certain parts of the park. A few have been placed in locations where you really would have to be in line for a specific attraction to access the fountain, but you should still be able to find an available one in each park regardless. The park maps do not currently reflect these locations, so you will want to ask a cast member where to find the nearest accessible fountain.
BONUS TIP
What if you decide to bring in a cooler, but don't want to carry it around all day long? In each park there are
lockers available to rent. They are at the front of each park after passing through the turnstiles, and there are a few locker sizes available. They are keyless and typically will cost you $15 or less for the day. The
Magic Kingdom Park and
EPCOT even have a few larger lockers available that are up to 17" x 22" x 26" (43 cm x 55 cm x 66 cm). Prices can change, so check first online for current rates.
BONUS TIP #2If you want to bring in a cooler it must be 24" x 15" x 18" (61 cm x 38 cm x 46 cm) or smaller.
Have a Magical Vacation!
Doug