  We determine a Simons' type formula for spacelike submanifolds within a broad
class of semiRiemannian warped products. This formula extends the Simons' type
formulas initially introduced by Nomizu and Smyth in 1969 for constant mean
curvature hypersurfaces in space forms. Furthermore, it incorporates the 2013
extension by Fetcu and Rosenberg for submanifolds with parallel mean curvature
vector field in product spaces. From a global standpoint, we establish that
compact spacelike hypersurfaces with parallel mean curvature and non-negative
sectional curvature are isoparametric hypersurfaces. This result constitutes a
generalization of the Riemannian case within space forms, as demonstrated by
Nomizu and Smyth. From a local standpoint, we extend analogous results
previously established by Asperti, Lobos and Mercuri for pseudo-parallel
immersions in Riemannian space forms. As consequences, we prove that any
semi-parallel spacelike hypersurface with zero mean curvature in the de Sitter
spacetime is totally geodesic. In contrast, there is no semi-parallel spacelike
hypersurface with zero mean curvature in the Einstein-de Sitter spacetime.
