Parallel pumping of spin waves in a ferromagnet revisited

•General theory for the parallel pumping threshold of spin-waves has been developed.•We show a possibility to use low-frequencies to study higher-frequency spin-waves.•The second-order resonance is achievable using a high-quality resonator.•The theory is applicable when ferromagnetic samples approach nano-sizes. A consistent theory of parametric resonance of spin-wave pairs in a ferromagnetically ordered system is developed for the case of parallel pumping. Exact equations for critical amplitudes at different frequencies of the pump field are obtained in the resonance approximation. The critical amplitudes for multiple frequencies of the pump field are calculated.