But pairity is just a question of sorting the countable numbers into sets of size two, and the more general form even of that is sorting into sets of size N. It's just as easy to say that the countable numbers are odd if there exists a way to sort them into sets of size three. So I'd argue the countable numbers are odd.

And you then you could retort with sets of size four, and I could use five, and then we can argue about whether we'll end up at the limit with more odd sets or even sets, and now we're arguing in circles. Reductio ad absurdum.