  We investigate plane-parallel wave metrics from the point of view of their
(Poisson-Lie) T-dualizability. For that purpose we reconstruct the metrics as
backgrounds of nonlinear sigma models on Lie groups. For construction of dual
backgrounds we use Drinfel'd doubles obtained from the isometry groups of the
metrics. We find dilaton fields that enable to satisfy the vanishing beta
equations for the duals of the homogenous plane-parallel wave metric. Torsion
potentials or B-fields, invariant w.r.t. the isometry group of Lobachevski
plane waves are obtained by the Drinfel'd double construction. We show that a
certain kind of plurality, different from the (atomic) Poisson-Lie T-plurality,
may exist in case that metrics admit several isometry subgroups having the
dimension of the Riemannian manifold. An example of that are two different
backgrounds dual to the homogenous plane-parallel wave metric.
