  We generalize a Bernstein-type result due to Albujer and Al\'ias, for maximal
surfaces in a curved Lorentzian product 3-manifold of the form $\Sigma_1\times
\mathbb{R}$, to higher dimension and codimension. We consider $M$ a complete
spacelike graphic submanifold with parallel mean curvature, defined by a map
$f: \Sigma_1\to \Sigma_2$ between two Riemannian manifolds $(\Sigma_1^m, g_1)$
and $(\Sigma^n_2, g_2)$ of sectional curvatures $K_1$ and $K_2$, respectively.
We take on $\Sigma_1\times \Sigma_2$ the pseudo-Riemannian product metric
$g_1-g_2$. Under the curvature conditions, $\mathrm{Ricci}_1 \geq 0$ and
$K_1\geq K_2$, we prove that, if the second fundamental form of $M$ satisfies
an integrability condition, then $M$ is totally geodesic, and it is a slice if
$\mathrm{Ricci}_1(p)>0$ at some point. For bounded $K_1$, $K_2$ and hyperbolic
angle $\theta$, we conclude $M$ must be maximal. If $M$ is a maximal surface
and $K_1\geq K_2^+$, we show $M$ is totally geodesic with no need for further
assumptions. Furthermore, $M$ is a slice if at some point $p\in \Sigma_1$,
$K_1(p)> 0$, and if $\Sigma_1$ is flat and $K_2<0$ at some point $f(p)$, then
the image of $f$ lies on a geodesic of $\Sigma_2$.
