  Topological excitations in many-body systems are one of the paradigmatic
cornerstones of modern condensed matter physics. In particular, parafermions
are elusive fractional excitations potentially emerging in fractional quantum
Hall-superconductor junctions, and represent one of the major milestones in
fractional quantum matter. Here, by using a combination of tensor network and
kernel polynomial techniques, we demonstrate the emergence of zero modes and
finite energy excitations in many-body parafermion chains. We show the
appearance of zero energy modes in the many-body spectral function at the edge
of a topological parafermion chain, their relation with the topological
degeneracy of the system, and we compare their physics with the Majorana bound
states of topological superconductors. We demonstrate the robustness of
parafermion topological modes with respect to a variety of perturbations, and
we show how weakly coupled parafermion chains give rise to in-gap excitations.
Our results exemplify the versatility of tensor network methods for studying
dynamical excitations of interacting parafermion chains, and highlight the
robustness of topological modes in parafermion models.
