Moiré materials provide a remarkably tunable platform for topological and strongly correlated quantum phases of matter. Very recently, the first Abelian fractional Chern insulators (FCIs) at zero magnetic field have been experimentally demonstrated, and it has been theoretically predicted that non-Abelian states with Majorana fermion excitations may be realized in the nearly dispersionless minibands of these systems. Here, we provide telltale evidence based on many-body exact diagonalization for the even more exotic possibility of moiré-based non-Abelian FCIs exhibiting Fibonacci parafermion excitations. In particular, we obtain low-energy quantum numbers, spectral flow, many-body Chern numbers, and entanglement spectra consistent with the Read–Rezayi parafermion phase in an exemplary moiré system with tunable quantum geometry. Our results hint towards the robustness of moiré-based parafermions and encourage the pursuit in moiré systems of these non-Abelian quasiparticles that are superior candidates for topological quantum computing.