  We present a joint experimental and theoretical study of parametric resonance
of spin wave eigenmodes in Ni$_{80}$Fe$_{20}$/Pt bilayer nanowires. Using
electrically detected magnetic resonance, we measure the spectrum of spin wave
eigenmodes in transversely magnetized nanowires and study parametric excitation
of these eigenmodes by a microwave magnetic field. We also develop an
analytical theory of spin wave eigenmodes and their parametric excitation in
the nanowire geometry that takes into account magnetic dilution at the nanowire
edges. We measure tuning of the parametric resonance threshold by antidamping
spin Hall torque from a direct current for the edge and bulk eigenmodes, which
allows us to independently evaluate frequency, damping and ellipticity of the
modes. We find good agreement between theory and experiment for parametric
resonance of the bulk eigenmodes but significant discrepancies arise for the
edge modes. The data reveals that ellipticity of the edge modes is
significantly lower than expected, which can be attributed to strong
modification of magnetism at the nanowire edges. Our work demonstrates that
parametric resonance of spin wave eigenmodes is a sensitive probe of magnetic
properties at edges of thin-film nanomagnets.
