  Given a class of compact spaces, we ask which groups can be maximal parabolic
subgroups of a relatively hyperbolic group whose boundary is in the class. We
investigate the class of 1-dimensional connected boundaries. We get that any
non-torsion infinite f.g. group is a maximal parabolic subgroup of some
relatively hyperbolic group with connected one-dimensional boundary without
global cut point. For boundaries homeomorphic to a Sierpinski carpet or a
2-sphere, the only maximal parabolic subgroups allowed are virtual surface
groups (hyperbolic, or virtually $\mathbb{Z} + \mathbb{Z}$).
