$R$ is reflexive as $(x+x)$ is even for any integer.
$R$ is symmetric as if $(x + y)$ is even $(y+x)$ is also even.
$R$ is transitive as if $(x + (y +z))$ is even, then $((x+y) + z)$ is also even.
So, $R$ is an equivalence relation.
For set of natural numbers, sum of even numbers always give even, sum of odd numbers always give even and sum of any even and any odd number always give odd. So, $R$ must have two equivalence classes -one for even and one for odd.
$\left\{\dots,-4,-2,0,2,4, \dots \right\}, \left\{\dots, -3,-1,1,3, \dots, \right\}$
C choice.