Classification of parabolic generating pairs of Kleinian groups with two parabolic generators

We give an alternative proof to Agol’s classification of parabolic generating pairs of non-free Kleinian groups generated by two parabolic transformations. As an application, we give a complete characterisation of epimorphims between $2$-bridge knot groups and a complete characterisation of degree one maps between the exteriors of hyperbolic $2$-bridge links.