Zero is neither even or odd, because zero is nothing, and you cannot make a picture of zero. This was the conclusion my 3rd graders came up with after working on the concept of even and odd numbers. We built pictures of numbers in the shape of 2x_ arrays. If the number had a tail it was odd, if the number had a partner it was even. Since zero can't be made into a picture, it is neither.
We started by drawing the numbers on graph paper. Then we made the numbers in 2x_ arrays with tiles. The next activity had students manipulating number puzzle pieces made from the 1 inch graph paper copied onto tag board. The students built number combinations, including 3+5, 2+4, 3+4. We paired pictures of the numbers to learn what happens when an odd number was paired with an odd number. The students generalized the ideas of odd+odd=even, because the "tails" fit together to form a partner. That even+even=even (all pictures have partners), and even+odd=odd, because the odd number will have a tail. After each activity, students wrote about the idea of even and odd in their math journals. To me, the personal writing is a critical step in student concept development. Each student had to explain why a number was even or odd in their own words. Their writing evolved as they did each activity.