  For a Riemannian manifold $M$, possibly with boundary, we consider the
Riemannian product $M\times\mathbb{R}^k$ with a smooth positive function that
weights the Riemannian measures. In this work we characterize parabolic
hypersurfaces with non-empty boundary and contained within certain regions of
$M\times\mathbb{R}^k$ with suitable weights. Our results include half-space and
Bernstein-type theorems in weighted cylinders. We also generalize to this
setting some classical properties about the confinement of a compact minimal
hypersurface to certain regions of Euclidean space according to the position of
its boundary. Finally, we show interesting situations where the statements are
applied, some of them in relation to the singularities of the mean curvature
flow.
