We prove that any degree d rational map having a parabolic fixed point of multiplier 1 with a fully invariant and simply connected immediate basin of attraction is mateable with the Hecke group Hd+1, with the mating realized by an algebraic correspondence. This solves the parabolic version of the Bullett-Freiberger Conjecture from 2003 on mateability between rational maps and Hecke groups. The proof is in two steps. The first is the construction of a pinched polynomial-like map which is a mating between a parabolic rational map and a parabolic circle map associated to the Hecke group. The second is lifting this pinched polynomial-like map to an algebraic correspondence via a suitable branched covering.
Submitted 20 July, 2024; originally announced July 2024.