Our aim is to give uniqueness results for entire solutions of certain family of PDEs of divergence form on a parabolic Riemannian manifold of arbitrary dimension. Each equation is the minimal hypersurface equation on certain warped product ambient space. These equations appear from a natural variational problem of geometric interest. Combining geometrical and analytical tools it is presented a technical result from which it is given new Moser's weak Bernstein theorems on parabolic manifolds.