  Parafermion zero modes are generalizations of Majorana modes that underlie
comparatively rich non-Abelian-anyon properties. We introduce exact mappings
that connect parafermion chains, which can emerge in two-dimensional
fractionalized media, to strictly one-dimensional fermionic systems. In
particular, we show that parafermion zero modes in the former setting translate
into 'symmetry-enriched Majorana modes' that intertwine with a bulk order
parameter---yielding braiding and fusion properties that are impossible in
standard Majorana platforms. Fusion characteristics of symmetry-enriched
Majorana modes are directly inherited from the associated parafermion setup and
can be probed via two kinds of anomalous pumping cycles that we construct. Most
notably, our mappings relate $\mathbb{Z}_4$ parafermions to conventional
electrons with time-reversal symmetry. In this case, one of our pumping
protocols entails fairly minimal experimental requirements: Cycling a weakly
correlated wire between a trivial phase and time-reversal-invariant topological
superconducting state produces an edge magnetization with quadrupled
periodicity. Our work highlights new avenues for exploring 'beyond-Majorana'
physics in experimentally relevant one-dimensional electronic platforms.
