There are many examples where parity matters a lot. Two classical examples from number theory:
- odd [perfect numbers](https://en.wikipedia.org/wiki/Perfect_number) are not expected to exist;
- even natural numbers greater than 2 are assumed to be the sum of two primes, according to [Goldbach's conjecture](https://en.wikipedia.org/wiki/Goldbach%27s_conjecture).

In a different field, [Sharkovskii ordering](https://en.wikipedia.org/wiki/Sharkovskii%27s_theorem) tells us that even periods are much more common among the set of real continuous functions.