Parity Addition is Associative/Proof 1

Proof
From Isomorphism between Ring of Integers Modulo 2 and Parity Ring:
 * $\left({\left\{ {\text{even}, \text{odd} }\right\}, +, \times}\right)$ is isomorphic with $\left({\Z_2, +_2, \times_2}\right)$

the ring of integers modulo $2$.

The result follows from:
 * Modulo Addition is Associative

and:
 * Isomorphism Preserves Associativity.