  We study a tight binding model of $\mathbb{Z}_3$-Fock parafermions with
single-particle and pair-hopping terms. The phase diagram has four different
phases: a gapped phase, a gapless phase with central charge $c=2$, and two
gapless phases with central charge $c=1$. We characterise each phase by
analysing the energy gap, entanglement entropy and different correlation
functions. The numerical simulations are complemented by analytical arguments.
