Seeking proofs that depend on the notion of even-vs-odd parity to prove their points The notion of the parity is very important in a variety of branches of mathematics. Specifically, I am looking for proofs that use parity in the even-vs-odd sense to prove their points.
For example, it is instructive to show that $\sqrt{2}$ is irrational by contradiction and assuming there exist relatively prime integers $m,n$ where $n \neq 0$ such that $\frac{m^2}{n^2} = 2$. By assuming that $m^2$ is even, you can arrive at a contradiction. A similar argument applies when assuming that $m^2$ is odd$.
Another example is showing that the alternating group of even permutations ($A_n$ within $S_n$) is a subgroup under composition of permutations. That claim can be extended to show that the odd permutations of $S_n$ do not form a subgroup.
I'm not looking for explanations of even/odd functions or how to partition $n$ into distinct odd/even parts, but rather my question is:

What proofs exists that depend on the notion of even-vs-odd parity to prove their points?

Ideally an answer will supply a description of the theorem/proof or a reference or a full proof itself.
 A: The parity of the order of a finite group plays an enormous role in finite group theory!
The Feit-Thompson Theorem settled Burnside's conjecture that every non-Abelian simple group has even order. 
There were numerous stepping-stones along the way, and a big part of the classification of finite simple groups was devoted to studying how involutions (non-identity elements of order two, guaranteed to exist in groups of even order, by Cauchy's theorem) affect the structure of a group; in particular looking at what's possible for the centralizer of an involution (apparently this is a consequence of the Brauer-Fowler Theorem, although I don't see the connection). 
One such stepping stone was Suzuki's CA paper, which studied groups (now called CA groups) in which the centralizer of any non-identity element is Abelian. In this paper, Suzuki showed that any CA group with odd order is solvable, and hence not simple.
A: In even-dimensional Euclidean spaces, spherical waves have trailing edges; in odd-dimensional spaces they don't.
A: The case of even perfect numbers is settled since long ($P=M(M+1)/2$ for $M$ a Mersenne prime). That of odd perfect numbers is still open.

The diophantine equation
$$x^p+y^p=z^p$$ where $p$ is a prime only has solutions for even $p$.
A: Euler, in his proof that all even perfect numbers have the form given by Euclid, used a very slick parity (odd/even) argument.
Also, parity is crucial in the canonical solution to the problem of generating Pythagorean triples.
Also, by a parity argument, it is proved that the number of sides of a golygon is a multiple of 8.
Here is the link to the Wikipedia article on golygons:
https://en.wikipedia.org/wiki/Golygon
It’s worth noting that in Language Arts, opposite parities can co-exist. That is, certain expressions, like ‘to dust’ possess antonymous meanings. Such expressions are called auto-antonyms. Here is the link to the Wikipedia article on this topic:
https://en.wikipedia.org/wiki/Auto-antonym#:~:text=An%20auto%2Dantonym%20or%20autantonym,is%20the%20reverse%20of%20another.&text=This%20phenomenon%20is%20called%20enantiosemy,%22)%2C%20antilogy%20or%20autantonymy.
