  In this paper we study analytic families of degree 2 parabolic-like mappings
(as we defined in arXiv:1111.7150). We prove that the corresponding family of
hybrid conjugacies induces a continuous map, which associates to each parameter
the multiplier of the fixed point of the hybrid equivalent rational map P_A. We
prove that, under suitable conditions, this map restricts to a ramified
covering from the connectedness locus of the family of parabolic-like maps to
the connectedness locus M_1 \ {1}.
