Covering of surfaces parametrized without projective base points

We prove that every affine rational surface, parametrized by means of an affine rational parametrization without projective base points, can be covered by at most three parametrizations. Moreover, we give explicit formulas for computing the coverings. We provide two different approaches: either covering the surface with a surface parametrization plus a curve parametrization plus a point, or with the original parametrization plus two surface reparametrizations of it.