  Parafermions are emergent excitations which generalize Majorana fermions and
are potentially relevant to topological quantum computation. Using the concept
of Fock parafermions, we present a mapping between lattice $\mathbb{Z}_4$
parafermions and lattice spin-$1/2$ fermions which preserves the locality of
operators with $\mathbb{Z}_4$ symmetry. Based on this mapping, we construct an
exactly solvable, local, and interacting one-dimensional fermionic Hamiltonian
which hosts zero-energy modes obeying parafermionic algebra. We numerically
show that this parafermionic phase remains stable in a wide range of
parameters, and discuss its signatures in the fermionic spectral function.
