Find irreducible components of $V(y − x^2, y^2 − y + z^2)$

Question

Let $F$ be an algebraically closed field. Find irreducible components of $X=V(y − x^2, y^2 − y + z^2)$ in $F^3$.

I think the variety itself is irreducible, but I couldn't prove that. I normally use the following approaches to check whether the variety is irreducible.

1. If I had $x - y^2$ as a generator instead of $y-x^2$ then the variety is a graph of a polynomial map (e.g $f: X \rightarrow F, X=V(y^2-y+z^2)$ which is irreducible).
2. parameterization
3. high school algebra (solving for $y − x^2=0$ and $y^2 − y + z^2=0$)
4. Show that $I(X)$ prime or $A(X)$ is isomorphic to an integral domain.