  Recent experiments brought Majorana particles closer to reality and raised
interest in realizing $\mathbb{Z}_k$ parafermions for general value of $k$ in
hybrid structures. Of particular interest is the prospect of realizing a two
dimensional system with the edge states described by $\mathbb{Z}_k$ parafermion
conformal field theory (CFT) by coupling one dimensional hybrid systems.
However in order to understand the effective perturbation due to local
inter-wire coupling, it is crucial to relate short distance ultra-violet (UV)
degrees of freedom to primary fields of the CFT describing long distance
infra-red (IR) physics. Here we consider two recent proposals for realizing a
one dimensional system described by $\mathbb{Z}_k$ parafermion CFT upon tuning
to quantum critical points: a parafermion chain and a pair of $\nu=2/k$ quantum
Hall chiral edge states under uniform backscattering and pairing, where the
latter is represented by self-dual sine-Gordon model. We explicitly derive the
relation between distinct UV degrees of freedom relevant for each proposals and
the primary fields of $\mathbb{Z}_3$ parafermion CFT. Our results point to a
marked difference in the effect of local coupling between wires represented by
each proposals, regarding robustness of the edge states in the resulting two
dimensional systems.
