  We consider a semiconducting nanowire with Rashba spin-orbit interaction
subjected to a magnetic field and in the presence of strong electron-electron
interactions. When the ratio between Fermi and Rashba momenta is tuned to
$1/2$, two competing resonant multi-particle scattering processes are present
simultaneously and the interplay between them brings the system into a gapless
critical parafermion phase. This critical phase is described by a self-dual
sine-Gordon model, which we are able to map explicitly onto the low-energy
sector of the $\mathbb{Z}_4$ parafermion clock chain model. Finally, we show
that by alternating regions in which only one of these two processes is present
one can generate localized zero-energy parafermion bound states.
