  The parametrically driven, damped nonlinear Schr\"odinger equation has two
cn- and two dn-wave solutions. We show that one pair of the cn and dn solutions
is unstable for any combination of the driver's strength, dissipation
coefficient and spatial period of the wave; this instability is against
periodic perturbations. The second dn-wave solution is shown to be unstable
against antiperiodic perturbations --- in a certain region of the parameter
space. We also consider quasiperiodic perturbations with long modulation
wavelength, in the limit where the driving strength is only weakly exceeding
the damping coefficient.
