Parity Addition is Associative/Proof 1

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

the ring of integers modulo $2$.

The result follows from:
 * Modulo Addition is Associative

and:
 * Isomorphism Preserves Associativity.