Is the function $z=x^{3}y$ even or odd? I just wanted to know because I am trying to calculate $\int_{-1}^{1}\int_{-1}^{1}x^{3}ydxdy$.
It is even because $(-x)^3(-y)=x^3y$.

Despite that, this is unnecessary for calculating your integral, where you can use  that $x^3$ and $y$ are odd, which results in the value $0$.
Parity is not defined for multivariate functions.

In your case, by Fubini

$$\iint f(x)\,g(x)\,dx\,dy=\int f(x)\,dx\int g(y)\,dy$$ and it suffices that any of the simple integrals vanishes.