  Non-Abelian anyons have garnered extensive attention for obeying exotic
non-Abelian statistics and having potential applications to fault-tolerant
quantum computing. While the prior research has predominantly focused on
non-Abelian statistics without the necessity of symmetry protection, recent
progresses have shown that symmetries can play essential roles and bring a
notion of the symmetry-protected non-Abelian (SPNA) statistics. In this work,
we extend the concept of SPNA statistics to strongly-correlated systems which
host parafermion zero modes (PZMs). This study involves a few fundamental
results proved here. First, we unveil a generic unitary symmetry mechanism that
protects PZMs from local couplings. Then, with this symmetry protection, the
PZMs can be categorized into two nontrivial sectors, each maintaining its own
parity conservation, even though the whole system cannot be dismantled into
separate subsystems due to nonlinear interactions. Finally, by leveraging the
parity conservation of each sector and the general properties of the effective
braiding Hamiltonian, we prove rigorously that the PZMs intrinsically obey SPNA
statistics. To further confirm the results, we derive the braiding matrix at a
tri-junction. We also propose a correlated quantum nanowire model that
accommodates a pair of PZMs protected by mirror symmetry and satisfying the
generic theory. This work shows a broad spectrum of strongly-correlated systems
capable of hosting fractional SPNA quasiparticles and enriches our
comprehension of fundamental quantum statistics linked to the symmetries that
govern the exchange dynamics.
