Show the ideal $(x^2-y,x^3-z)$ is prime using the definition

How can we show that the ideal $\mathfrak{p}:=(x^2-y,x^3-z) \subset k[x,y,z]$ is prime using the definition of prime ideals in commutative rings?

To show the ideal is prime one defines a map $k[x,y,z] \to k[t]$ given by $f(x,y,x)\mapsto f(t,t^2,t^3)$, and shows that $k[x,y,z]/\mathfrak{p} \cong k[t]$ is PID, and so domain. Thus, $\mathfrak{p}$ is prime.

I want to see how we can prove this by showing $fg \in \mathfrak{p} \Longrightarrow f \in \mathfrak{p}\text{ or } g \in \mathfrak{p}$. I was thinking to say if $f$ and $g$ are not in the ideal, then there exists $t_1$ and $t_2$ such that $f(t_1,t_1^2,t_1^3) \ne 0$ and $g(t_2,t_2^2,t_2^3) \ne 0$. But how can this be used to show the existence of $t$ such that $fg(t,t^2,t^3)\ne 0$?