Why? Seriously, though: what do you mean by parity (EDIT: I mean even-ness) of a set? Almost assuredly, something like “it can be partitioned into two disjoint subsets that can be put into bijection”; but then you've just defined even-ness of cardinals (which are sets after all), and all infinite cardinals are even (in fact, satisfy 2κ = κ; EDIT: and odd; they also satisfy 2κ + 1 = κ).
You read my mind for "parity of a set". So Q.E.D.: parity is only applicable to reason about numbers (and infinity is not a number). See also surreal numbers.
It's a property of _integers_, or any system with a natural homorphism into the integers mod 2 under addition. It's certainly not a property of numbers in general. Is pi even or odd? How about sqrt(2) + i*e?
Why? Seriously, though: what do you mean by parity (EDIT: I mean even-ness) of a set? Almost assuredly, something like “it can be partitioned into two disjoint subsets that can be put into bijection”; but then you've just defined even-ness of cardinals (which are sets after all), and all infinite cardinals are even (in fact, satisfy 2κ = κ; EDIT: and odd; they also satisfy 2κ + 1 = κ).