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.
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