  We study charge transport through a floating mesoscopic superconductor
coupled to counterpropagating fractional quantum Hall edges at filling fraction
$\nu=2/3$. We consider a superconducting island with finite charging energy and
investigate its effect on transport through the device. We calculate
conductance through such a system as a function of temperature and gate voltage
applied to the superconducting island. We show that transport is strongly
affected by the presence of parafermionic zero modes, leading at zero
temperature to a zero-bias conductance quantized in units of $\nu e^2/h$
independent of the applied gate voltage.
