A whole number is even if it is divisible by 2.  Equivalently (since sometimes there is confusion with 0 and the word ‘divisible’), a whole number is even if it can be written as two times some whole number.  E.g., 22 is even because it can be written as 2 time somes whole number (namely 11), and 7 is NOT even because it can’t be written as 2 times some whole number.
That’s all there is to it.  SO, to ask “Is o even?” is to ask “Can it be written as 2 times some other whole number?”  AND, since that answer is yes (2 x 0 = 0), we say that 0 is even.
If we’re careful (and have already defined even), it’s OK to say a whole number is odd if it isn’t even.  We could – maybe should – get more technical, but I won’t.  A whole number is either even or odd, never both.
SO, since 2 IS even, it is NOT odd.  And therefore it is NOT both (and not neither, for that matter. 🙂 ).

As long as we’re here, let’s quickly review positive and negative.  An integer is positive if it’s greater than 0, and it’s negative if it’s less than 0.  Since 0 is not greater than or less than itself, it is NEITHER positive nor negative.

NOTE – and PREVIEWS 🙂  We have used terms like ‘integer’ and ‘whole number’.  Do you remember all the distinctions for the various sets of numbers we use?   That will show up as a Math Tidbit very soon.  So might the discussion of the why we don’t divide by 0.