The offical line is that no outside food to be eaten in the parks. If you bring your lunch with you, you are supposed to eat it at the picnic tables outside. Partly I think because they feel if its an inconvience you won't do it, but mostly because then people take up seats for the people who pay to eat there. Believe me there is nothing worse than buying yourself a meal and not being able to find a table (especially inside on a hot day) and seeing tables full of people eating their food brought from home.
Snacks are not a problem though. They used to be a lot pickier about them, but now they don't really say much. If you bring in a cooler (anything bigger than a small over the shoulder type cooler) they will not let you in the park with it. Over by the picnic tables they have a bunch of lockers that are big enough for coolers and they will direct you there to store it (or back at your car).
Glass bottles are not allowed in if they see them.