  We devise local lattice models whose ground states are model fractional Chern
insulators---Abelian and non-Abelian topologically ordered states characterized
by exact ground state degeneracies at any finite size and infinite entanglement
gaps. Most saliently, we construct exact parent Hamiltonians for two distinct
families of bosonic lattice generalizations of the $\mathcal{Z}_k$ parafermion
quantum Hall states: (i) color-entangled fractional Chern insulators at band
filling fractions $\nu=k/(\mathcal{C}+1)$ and (ii) nematic states at $\nu=k/2$,
where $\mathcal{C}$ is the Chern number of the lowest band in our models. In
spite of a fluctuating Berry curvature, our construction is partially
frustration free: the ground states reside entirely within the lowest band and
exactly minimize a local $(k+1)$-body repulsion term by term. In addition to
providing the first known models hosting intriguing states such as higher Chern
number generalizations of the Fibonacci anyon quantum Hall states, the
remarkable stability and finite-size properties make our models particularly
well-suited for the study of novel phenomena involving e.g. twist defects and
proximity induced superconductivity, as well as being a guide for designing
experiments.
