I think I have an answer but I'm looking for some verification.
Since $x - yz = 0$, then I can rewrite my problem as $V(x - yz, yz^2 - y^2)$.
This is equal to $V(x - yz, y) \cup V(x - yz, z^2 - y)$.
Then $V(x - yz, y) = V(x, y)$ is irreducible.
Also, $V(x - yz, z^2 - y) = V(x - z^3, z^2 - y)$ is also irreducible, since the corresponding coordinate ring will be $\mathbb{C}[z],$ which is an integral domain.
So the irreducible components are $V(x, y)$ and $V(x - z^3, z^2 - y).$