If I were trying to explain this to someone who is just learning about odd and even then I might argue this way.
Suppose I have a bag of gum drops and I want to share them with a friend. If I can divide them so that we each have the same number of gum drops then the number of gum drops is even. If not then the number of gum drops I have is odd. Now suppose I have two bags of gum drops, each with an odd number of gum drops, and I want to share them with a friend. I start with the first bag and divide them between the two of us. There is one left over. Then I do the same with the second bag and there is also one left over. I give one of the left over gum drops to my friend and keep the other so I have successfully shared the gum drops in the two bags. Thus the total number of gum drops, the sum of two odd numbers, is even.
Said more simply.
An odd number is one more than an even number so the sum of two odd numbers is the sum of two even numbers and two more. Since the sum of two evens is even adding two more again gives an even number.
And said algebraically.
Let p and q be odd numbers than there are integers n and k so that p = 2n + 1 and q = 2k + 1. Hence
p + q = 2n + 1 + 2k + 1 = 2n + 2k + 2 = 2(n + k + 1)
Hence p + q is even.
I hope this helps,