From the Disney World FAQs (reiterating what others have said):
Items that you may not bring into the Theme Parks include, but are not limited to:
Items with wheels, such as wagons, skateboards, scooters, inline skates, shoes with built-in wheels, two-wheeled or three-wheeled conveyances, strollers larger than 36" x 52", suitcases, coolers, or backpacks with or without wheels larger than 24" long x 15" wide x 18" high (coolers required for medication may be stored in a locker or at Guest Relations), and any trailer-like object that is pushed or towed by an
ECV wheelchair or stroller
Alcoholic beverages
Folding chairs
Glass containers (excluding baby food jars and perfume bottles)
In Disney's Animal Kingdom® Theme Park (for the safety of the wildlife), balloons, straws and drink lids are not permitted.
I think it's safe to say "unofficially" it's discouraged since Disney anticipates parkgoers to purchase their food within the park and obviously does not set up picnic areas and such to encourage it. I'm sure they don't want to face the backlash of issues related to not bringing in food such as how do you enforce it, who draws the line at what is acceptable, etc.
BUT...I agree with TheRustyScupper on this one for the most part