Fibonacci anyons provide a universal platform for topological quantum computation, and emerge as low-energy excitations in the Z₃ Read-Rezayi phase in the fractional quantum Hall effect. However, realistic microscopic realizations of this phase in the absence of a magnetic field have remained elusive. We study a model of periodically modulated Bernal bilayer graphene with gate-screened Coulomb interactions. Using the recently developed target-phase optimization method in conjunction with band-projected exact diagonalization, we identify at filling ν=3/5 a region of parameter space whose ground state is consistent with a Read-Rezayi fractional Chern insulator. The partially filled band from which it arises is a part of a two-band complex which mimics geometric aspects of the lowest and first Landau levels, with the ground state at ν=1/2 consistent with the Moore-Read state. Our results suggest that modulated Bernal graphene can realize delicate non-Abelian fractional quantum Hall states at zero magnetic field, while demonstrating target-phase optimization as a practical route to discovering such phases in realistic, high-dimensional microscopic models.