  Using the decomposition of rational conformal filed theory characters for the
${\mathbb Z}_k$ parafermion quantum Hall droplets for general $k=2,3, \ldots$,
we derive analytically the full modular $S$ matrix for these states, including
the $\widehat{u(1)}$ parts corresponding to the charged sector of the full
conformal field theory and the neutral parafermion contributions corresponding
to the diagonal affine coset models. This precise neutral-part parafermion $S$
matrix is derived from the explicit relations between the coset matrix and
those for the numerator and denominator of the coset and the latter is
expressed in compact form due to the level--rank duality between the affine Lie
algebras $\widehat{su(k)_2}$ and $\widehat{su(2)_k}$. The exact results
obtained for the $S$ matrix elements are expected to play an important role for
identifying interference patterns of fractional quantum Hall states in
Fabry-P\'erot interferometers which can be used to distinguish between Abelian
and non-Abelian statistics of quasiparticles localized in the bulk of
fractional quantum Hall droplets as well as for nondestructive interference
measurement of Fibonacci anyons which can be used for universal topological
quantum computation.
