I have the surface defined by $z=y^2-x^2$ for $z\ge0$. A possible parametrization for the surface should be $\vec{r}(u,v)=\left(\begin{matrix}u\sinh(v)\\u\cosh(v)\\u^2\end{matrix}\right);u,v\in\mathbb{R}$.
If I look at the contour line where $z=0$ it should give me $y=\pm|x|$. However, from the parametrization, I get $z=0\iff u=0\Rightarrow x=0,y=0$. Can someone explain what is going on here?
The level curve for $z=0$ is $y^2-x^2=0$. There’s no parameterization of the form $(a\sinh t,b\cosh t)$ for this degenerate hyperbola. That only works for nondegenerate hyperbolas, which are all of the other level curves of the surface. So, this parameterization of the surface can’t work for $z=0$, as you’ve discovered.
If you need a parameterization that works for $z=0$, try taking vertical slices through the surface instead and parameterizing the resulting family of parabolas.