  Using the conformal field theory approach to the thermoelectric
characteristics of fractional quantum Hall states, previously developed in
Nucl. Phys. B 894 (2015) 284, we show that the thermoelectric power factor of
Coulomb-blockaded islands, realized by point contacts in Fabry--P\'erot
interferometers in the $\mathbb{Z}_k$ parafermion Hall states, could give
reliable signatures for distinguishing the topological orders of different
quantum Hall states having identical electric properties. For example, while
the conductance peak patterns in the Coulomb blockade regime for such states
are practically indistinguishable for $v_n \ll v_c$ even at finite temperature,
where $v_n$ and $v_c$ are the Fermi velocities of the neutral and charged modes
respectively, the power factors $\mathcal{P}_T$ of the corresponding states are
much more sensitive to the neutral modes. In particular, the smaller
$r=v_n/v_c$ the bigger the asymmetries in the power factor which combined with
the thermal broadening of the conductance peaks due to the neutral modes'
multiplicities could give us the ultimate tool to figure out which of the
competing quantum Hall universality classes are indeed realized in the
experiments. We give a complete description of the power factor profiles in the
$\mathbb{Z}_3$ and $\mathbb{Z}_4$ parafermion states with arbitrary number of
quasiparticles localized in the bulk which could be useful for comparison with
the experiments.
