Answer
\[
r^{2} \cos 2 \theta=z, r \cos \theta=x, r \sin \theta=y
\]
Work Step by Step
\[
\begin{array}{l}
\qquad r \cos \theta=x, r \sin \theta=y \\
\Rightarrow z=y^{2} x^{2}=(r \cos \theta)^{2}(r \sin \theta)^{2}=r^{2}\left(\cos ^{2} \theta \sin ^{2} \theta\right)=r^{2} \cos 2 \theta \quad\left(\because \cos 2 \theta=\cos ^{2} \theta \sin ^{2} \theta\right)
\end{array}
\]
So, the surface can be represented parametrically as:
\[
r^{2} \cos 2 \theta=z, r \cos \theta=x, r \sin \theta=y
\]
