It's not a number. It's a process of generating an unbounded series of numbers. Then, what kind of infinity do are we talking about. Monotonic number series interleave odd and even numbers. The series of all positive odd numbers will by definition generate odd numbers. And so on.
It’s a cardinal number: the size of sets. Adding a single new element to an infinite set doesn’t change its size, so adding 1 couldn’t change an odd/even property of that cardinal number.
But there are also infinities as ordinal numbers, and for those there is a well-defined notion of odd/even, based on their total ordering.
