  We classify invariant probability measures for non-elementary groups of
automorphisms, on any compact K\"ahler surface X, under the assumption that the
group contains a so-called "parabolic automorphism". We also prove that except
in certain rigid situations known as Kummer examples, there are only finitely
many invariant, ergodic, probability measures with a Zariski dense support. If
X is a K3 or Enriques surface, and the group does not preserve any algebraic
subset, this leads to a complete description of orbit closures.
