Consider $I=(x^2 - y^3, y^2 - z^3)\subset k[x,y,z]$ as an ideal with $k$ a field. 

$\textbf{Q:}$ I am Looking for alternative argument to conclude $I$ is prime ideal. It is clear that I can use parametrization $k[x,y,z]\to k[t]$ by parametrization $(x,y,z)\to (t^9, t^6, t^4)$ and argue this descends to a monomorphism under quotient $k[x,y,z]/I$. Note this map is clearly not surjection. Hence this is not isomorphic to $A^1\_k$ and this is already indicated by singularity at $(0,0,0)$. Can I conclude primeness of ideal $I$ by some other better argument? Say intersection,grobner basis,...