  Parafermions are emergent excitations that generalize Majorana fermions and
can also realize topological order. In this paper we present a non-trivial and
quasi-exactly-solvable model for a chain of parafermions in a topological
phase. We compute and characterize the ground-state wave-functions, which are
matrix-product states and have a particularly elegant interpretation in terms
of Fock parafermions, reflecting the factorized nature of the ground states.
Using these wavefunctions, we demonstrate analytically several signatures of
topological order. Our study provides a starting point for the non-approximate
study of topological one-dimensional parafermionic chains with
spatial-inversion and time-reversal symmetry in the absence of strong edge
modes.
