Optical parametric oscillation (OPO) using the third-order nonlinearity (chi((3))) in integrated photonics platforms is an emerging approach for coherent light generation, and has shown great promise in achieving broad spectral coverage with small device footprints and at low pump powers. However, current chi((3)) nanophotonic OPO devices use pump, signal, and idler modes of the same transverse spatial mode family. As a result, such single-mode-family OPO (sOPO) is inherently sensitive in dispersion and can be challenging to scalably fabricate and implement. In this work, a novel scheme is proposed using different families of transverse spatial modes for pump, signal, and idler, which is termed as hybrid-mode-family OPO (hOPO). The hOPO shows unprecedented robustness in dispersion versus device geometry, pump frequency, and temperature. Moreover, the hOPO is capable of generating a few milliwatts of output signal power with a conversion efficiency approximate to 8% of and without competitive processes. This hOPO scheme is an important counterpoint to the existing sOPO approach, and is particularly promising as a robust method to generate coherent on-chip visible and infrared light sources.