Motivated by recent observations of fractional Chern insulators (FCIs) in the vicinity of superconducting (SC) phases (e.g. $\text{MoTe}_2$), we study fractional quantum (anomalous) Hall-superconductor heterostructures in the presence of U(1) order-parameter fluctuations and particularly focus on the case of $\nu=2/3$ quantum Hall states leading to $Z_3$ parafermions. While the emergence of $Z_3$ parafermions has already been demonstrated in the spin-unpolarized Halperin State, we demonstrate how disorder can also stabilize $Z_3$ parafermions in the spin-polarized $\nu=2/3$ state, which is expected to be present in $\text{MoTe}_2$.
Due to a broad phase transition between the FCI and the superconducting state, we further study the effect of phase fluctuations on the $Z_3$ parafermions, using field-theoretical methods as well as the density-matrix renormalization group. We find a rich phase diagram composed of Mott insulating phases and two distinct Luttinger liquids, whose fundamental excitations carry charges 2e and 2e/3, respectively. In agreement with analytical considerations using conformal field theory, we numerically find transitions of Berezinskii-Kosterlitz-Thouless (BKT) type as well as a continuous $Z_3\times\text{U(1)}$ second-order phase transition characterized by central charge c = 9/5.