I think the reals are also even: If x is rational pair it as you would in the rational case (which we assume is even - I haven't proven this). Otherwise pair it to -x, and thus the reals are even.

Being "even" seems like a much more interesting (and simpler) property of a set. I don't see what use there could be to know that you could pair things off, with one element left over. When you extend the notion you do have to decide what to preserve, but to me parity is much more about divisibility and symmetry than it is about reminader. I agree that it's arbitrary, though less arbitrary than the odd definition.