Evenness is a more natural condition, so to speak, in that it has a simple definition and is easy to generalize. Having defined an even number, if an integer isn't even, it's odd.
To get a feel for why this is convenient, consider that you can generalize by replacing "multiples of 2" with "multiples of n". Then, instead of splitting everything into two sets (even/odd), we can naturally split the integers into n sets called equivalence classes modulo n. For n=10, these would be "multiples of 10", "numbers whose remainder after dividing by 10 is 1", "numbers whose remainder after dividing by 10 is 2", and so on. Seen this way, you may find it less arbitrary now.
To get a feel for why this is convenient, consider that you can generalize by replacing "multiples of 2" with "multiples of n". Then, instead of splitting everything into two sets (even/odd), we can naturally split the integers into n sets called equivalence classes modulo n. For n=10, these would be "multiples of 10", "numbers whose remainder after dividing by 10 is 1", "numbers whose remainder after dividing by 10 is 2", and so on. Seen this way, you may find it less arbitrary now.