Parabolic bifurcation loci in the spaces of rational functions

We give a geometric description of the parabolic bifurcation locus in the space $\operatorname{Rat}_d$ of all rational functions on $\mathbb{P}^1$ of degree $d>1$, generalizing the study by Morton and Vivaldi in the case of monic polynomials. The results are new even for quadratic rational functions.