how do you find a parametrization of the cylinder $x^2+y^2=1$ the lies between the planes $z=-1$ and $z=x^2+y^2$

Question

Since we are dealing with a cylinder, I thought it would be wise to use the cylindrical coordinate system.

Therefore

$x=rcos\theta$

$y=rsin\theta$

$z = x^2+y^2=r^2$

Therefore my final answer would be:

$r(r,\theta, z) = (rcos\theta, rsin\theta, r^2) $

But that seems like a parmeterization of just the cylinder, without the intersection of the plane and the paraboloid.

I would appreciate any input.