  Parafermions are anyons with the potential for realizing non-local qubits
that are resilient to local perturbations. Compared to Majorana zero modes,
braiding of parafermions implements an extended set of topologically protected
quantum gates. This, however, comes at the price that parafermionic zero modes
can not be realized in the absence of strong interactions whose theoretical
description is challenging. In the present work, we construct a simple lattice
model for interacting spinful electrons with parafermionic zero energy modes.
The explicit microscopic nature of the considered model highlights new
realization avenues for these exotic excitations in recently fabricated quantum
dot arrays. By density matrix renormalization group calculations, we identify a
broad range of parameters, with well-localized zero modes, whose parafermionic
nature is substantiated by their unique $8\pi$ periodic Josephson spectrum.
