The Modular Mandelbrot set is the connectedness locus of a family of (2:2) correspondences (introduced by Bullett and Penrose in 1994). We show that these correspondences are matings between the modular group and the family of quadratic rational maps P_A(z)=z+1/z+A, and that there exists a dynamical homeomorphism between the modular Mandelbrot set and the parabolic Mandelbrot set (this last being the connectedness locus of the family P_A(z), and is itself homeomorphic to the classical Mandelbrot set by a result of Petersen and Roesch). The talk is based on joint work with Shaun Bullett.