Associated prime of certain tensor product

Question

Let $A$ be a Noetherian ring and $p\subset A$ a prime ideal. Is $p\otimes_A A/p$ a torsion-free $A/p$-module? The case which I'm interested in is $A:=k[X,Y,Z]/(X^2-YZ)$ and $p:=(x,y)$, where $k$ is a field and $x,y,z$ are the images of $X,Y,Z$ in $A$. I know that the ideal $(x,y,z)$ is associated to the $A$-module $p/p^2$. I want to show that it is not associated to $p\otimes_A A/p$. But I'm not sure whether what I'm trying to prove is correct or not.