  The parametrically driven damped nonlinear Schr\"odinger equation serves as
an amplitude equation for a variety of resonantly forced oscillatory systems on
the plane. In this note, we consider its nodal soliton solutions. We show that
although the nodal solitons are stable against radially-symmetric perturbations
for sufficiently large damping coefficients, they are always unstable to
azimuthal perturbations. The corresponding break-up scenarios are studied using
direct numerical simulations. Typically, the nodal solutions break into
symmetric "necklaces" of stable nodeless solitons.
