We study non-compact complete spacelike hypersurfaces in generalized Robertson–Walker spacetimes of arbitrary dimension whose fiber is parabolic. Under boundedness assumptions on the warping function restricted on a spacelike hypersurface and on the hyperbolic angle of the hypersurface, we prove that a complete spacelike hypersurface is parabolic if the Riemannian universal covering of the fiber is so. As an application of this new technique, several uniqueness results on complete maximal spacelike hypersurfaces are obtained. Also, the corresponding Calabi–Bernstein problems are solved.