Hyperbolic geometry is not really geometry on a hyperboloid. It's geometry on an infinite surface of constant negative Gaussian curvature, something which cannot be represented even in 3D. You can model it using a sheet of a hyperboloid, but the metric you get isn't the normal 3D metric you'd intuitively expect.
Elliptic geometry is not the geometry on an ellipsoid either. While spherical geometry is what you get as geometry on the sphere, elliptic geometry is what you get from that if you identify antipodal pairs of points. It's the geometry on a surface of constant positive Gaussian curvature.
Just like the parabola is the singular limiting case between ellipse and hyperbola, the parabolic geometry is the limiting case between elliptic and hyperbolic geometry. And between constant positive and constant negative curvature, that limiting case is zero curvature.
Might be you could model parabolic geometry on a paraboloid using some strange metric, but why bother if you can have a flat plane using normal Euclidean metric, perfectly intuitive?
For a nice uniform way of looking at these different geometries, I suggest looking into Cayley-Klein metrics. Perspectives on Projective Geometry by Richter-Gebert has some nice chapters on this. Disclaimer: I'm working with that author, so I might be somewhat biased here.