  In this paper we provide several uniqueness and non-existence results for
complete parabolic constant mean curvature spacelike hypersurfaces in
Lorentzian warped products under appropriate geometric assumptions. As a
consequence of this parametric study, we obtain very general uniqueness and
non-existence results for a large family of uniformly elliptic EDP's, so
solving the Calabi-Bernstein problem in a wide family of spacetimes.
