  A surface in hyperbolic space $\h^3$ invariant by a group of parabolic
isometries is called a parabolic surface. In this paper we investigate
parabolic surfaces of $\h^3$ that satisfy a linear Weingarten relation of the
form $a\kappa_1+b\kappa_2=c$ or $aH+bK=c$, where $a,b,c\in \r$ and, as usual,
$\kappa_i$ are the principal curvatures, $H$ is the mean curvature and $K$ is
de Gaussian curvature. We classify all parabolic linear Weingarten surfaces in
hyperbolic space.
