In this paper, we introduce and characterize a class of
parabolically extended structures for relatively hyperbolic groups. A characterization of relative quasiconvexity with respect to
parabolically extended structures is obtained. The connection
between the existence of a minimal relatively hyperbolic structure
on a given group and the action on its Floyd boundary is examined. It is shown that Dunwoody's inaccessible group that has no minimal
relatively hyperbolic structure turns out to be acting
non-geometrically finitely on its Floyd boundary.