On rank two Kleinian groups with three parabolics

The free group of rank $2$ is the fundamental group of the genus $2$ handlebody $\mathcal{H}$. We study discrete representations of this group into $\mathsf{P}\mathsf{S}\mathsf{L}(2,\mathbb{C})$ so that three disjoint simple closed curves on the conformal boundary ${\mathrm{\partial }}_{\mathrm{\infty }}\mathcal{H}$ are sent to parabolic elements. We show that the only infinite covolume groups of this form are maximal cusp groups on the boundary of genus $2$ Schottky space. We also exhibit hitherto unexpected finite covolume groups which do not arise from Heegaard splitting presentations of tunnel number $1$ links.