  We study Poisson-Lie T-duality of the Wess-Zumino-Novikov-Witten (WZNW)
models which are obtained from a class of Drinfel'd doubles and its
generalization. In this case, the resultant WZNW models are known to be
classically self-dual under Poisson-Lie T-duality. We describe an explicit
construction of the associated currents, and discuss the conformal invariance
under this duality. In a concrete example of the SU(2) WZNW model, we find that
the self-duality is represented as a chiral automorphism of the
$\widehat{\mathfrak{su}}(2)$ affine Lie algebra, though the transformation of
the currents is non-local and non-linear. This classical automorphism can be
promoted to the quantum one through the parafermionic formulation of
$\widehat{\mathfrak{su}}(2)$, which in turn induces an isomorphism of the WZNW
model. We thus find a full quantum equivalence of the dual pair under
Poisson-Lie T-duality. The isomorphism is represented by a sign-change of a
chiral boson or the order-disorder duality of the parafermionic conformal field
theory as in Abelian T-duality on tori or in the mirror symmetry of the Gepner
model.
