  We construct a parabolic entire minimal graph $S$ over a finite topology
complete Riemannian surface $\Sigma$ of curvature $-1$ and infinite area (thus
of non-parabolic conformal type). The vertical projection of this graph yields
a harmonic diffeomorphism from $S$ onto $\Sigma$. The proof uses the theory of
divergence lines to construct minimal graphs.
  We also generalize a theorem of R. Schoen. Let $g_1$ and $g_2$ be two
complete metrics on a orientable surface $S$ with compact boundary and suppose
$$\int_{S_r^2}K_{g_2}^-d\sigma_{g_2}\le C\ln(2+r)$$ for some $C>0$ and all
$r>0$. If there is a harmonic diffeomorphism from $(S,g_1)$ to $(S,g_2)$, then
$(S,g_1)$ is parabolic.
