  We study optical parametric oscillations in Kerr-nonlinear microresonators,
revealing an intricate solution space -- parameterized by the pump-to-signal
conversion efficiency -- that arises from an interplay of nonlinear processes.
Using a three-mode approximation, we derive an efficiency-maximizing relation
between pump power and frequency mismatch. To move beyond a three-mode
approximation, a necessity for geometries such as integrated microring
resonators, we numerically simulate the Lugiato-Lefever Equation that accounts
for the full spectrum of nonlinearly-coupled resonator modes. We observe and
characterize two nonlinear phenomena linked to parametric oscillations in
multi-mode resonators: Mode competition and cross phase modulation-induced
modulation instability. Both processes may impact conversion efficiency.
Finally, we show how to increase the conversion efficiency by tuning the
microresonator loss rates. Our analysis will guide microresonator designs that
aim for high conversion efficiency and output power.
