Here are some such:
1. Euler, in his proof that all even perfect numbers have the form given by Euclid, used a very slick parity (odd/even) argument.
2. Parity is crucial in the canonical solution to the problem of generating Pythagorean triples.
3. By a parity argument, it is proved that the number of sides of a golygon is a multiple of 8.
[odd and even]
[oddness and evenness]