A non-elementary M\"obius group generated by two-parabolics is determined upto conjugation by one complex parameter and the parameter space has beenextensively studied. In this paper, we use the results of \cite{GW} to obtainan additional structure for the parameter space, which we term the {\sltwo-parabolic space}. This structure allows us to identify groups that containadditional conjugacy classes of primitive parabolics, which following\cite{Indra} we call {\sl parabolic dust groups}, non-free groups off the realaxis, and groups that are both parabolic dust and non-free; some of thesecontain $\mathbb{Z} \times \mathbb{Z}$ subgroups. The structure theorem alsoattaches additional geometric structure to discrete and non-discrete groupslying in given regions of the parameter space including a new explicitconstruction of some non-classical $\T$Schottky groups.
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