Finding parametric equations for curve of intersection I am having difficulty finding parametric equations of the curve of intersection of $z = x^2 - y^2$ and cylinder $x^2 + y^2 = 1$

I am aware that the first equation represents a hyperboloid, and the second equation represents a cylinder. I think for the second one I can just do $x = cos(t)$ and $y = sin(t)$. Not sure about the other one.
We have the two surfaces

$$
x^2-y^2 = z\\
x^2+y^2 = 1
$$

Solving for $x^2, y^2$ we have

$$
x^2 = \frac 12(1+z)\\
y^2 = \frac 12(1-z)
$$

so a parametric representation can be read as

$$
(x(s),y(s),z(s)) = \left(\pm\sqrt{\frac 12(1+s)},\pm\sqrt{\frac 12(1-s)},s\right)
$$

for $-1\le s \le 1$

Attached a plot with the four leafs
For the first one use that $$\cosh^2(t)-\sinh^2(t)=1$$
And since$$x^2=z+y^2$$ we get

$$z+2y^2=1$$ so....
$$z=1-2y^2$$