Yes, you can bring outside food into the parks. I have often brought snacks for my children with me through security and the park entrance turnstiles -- and many people with allergies or other food sensitivities bring supplementary food in with them.
But what you can't bring into the parks are large coolers. If you click
here, you'll find a list of Frequently Asked Disney parks questions. At the bottom of the list, you'll see the maximum dimensions of coolers allowed in the parks. Other items not allowed into the parks are coolers with wheels and glass containers. As long as you stay within these guidelines, you can enjoy your own food in the theme parks.
Also bear in mind that there is not much storage space in the theme parks. There are lockers available for rental, but they are not large, so whatever you bring with you, you will likely be carrying around.
Hope that helps.