  We explore the $\mathbb{Z}_{N}$ parafermionic clock-model generalisations of
the p-wave Majorana wire model. In particular we examine whether zero-mode
operators analogous to Majorana zero-modes can be found in these models when
one introduces chiral parameters to break time reversal symmetry. The existence
of such zero-modes implies $N$-fold degeneracies throughout the energy
spectrum. We address the question directly through these degeneracies by
characterising the entire energy spectrum using perturbation theory and exact
diagonalisation. We find that when $N$ is prime, and the length $L$ of the wire
is finite, the spectrum exhibits degeneracies up to a splitting that decays
exponentially with system size, for generic values of the chiral parameters.
However, at particular parameter values (resonance points), band crossings
appear in the unperturbed spectrum that typically result in a splitting of the
degeneracy at finite order. We find strong evidence that these preclude the
existence of strong zero-modes for generic values of the chiral parameters. In
particular we show that in the thermodynamic limit, the resonance points become
dense in the chiral parameter space. When $N$ is not prime, the situation is
qualitatively different, and degeneracies in the energy spectrum are split at
finite order in perturbation theory for generic parameter values, even when the
length of the wire $L$ is finite. Exceptions to these general findings can
occur at special "anti-resonant" points. Here the evidence points to the
existence of strong zero modes and, in the case of the achiral point of the the
$N=4$ model, we are able to construct these modes exactly.
