  Search for parafermions and Fibonacci anyons, which are excitations obeying
non-Abelian statistics, is driven both by the quest for deeper understanding of
nature and prospects for universal topological quantum computation. However,
physical systems that can host these exotic excitations are rare and hard to
realize in experiments. Here we study the domain walls and the edge states
formed in spin transitions in the fractional quantum Hall effect. Effective
theory approach and exact diagonalization in a disk and torus geometries proves
the existence of the counter-propagating edge modes with opposite spin
polarizations at the boundary between the two neighboring regions of the
two-dimensional electron liquid in spin-polarized and spin-unpolarized phases.
By analytical and numerical analysis, we argue that these systems can host
parafermions when coupled to an s-wave superconductor and are experimentally
feasible. We investigate settings based on $\nu=\frac{2}{3}$, $\nu=\frac{4}{3}$
and $\nu=\frac{5}{3}$ spin transitions and analyze spin-flipping interactions
that hybridize counter-propagating modes. Finally, we discuss spin-orbit
interactions of composite fermions.
