In doubly resonant optical parametric oscillators (DROPOs), it is possible to generate, enhance, and phase lock two frequencies at once. Following intracavity phase conditions, a complex tuning behavior of the signal and idler spectra takes place in DROPOs, cumulating into degeneracy with phase self-locking and coherent wavelength doubling. In this work, we identify group delay matching as the important parameter determining the global tuning behavior and demonstrate the key role of higher-order dispersion in the spectral dependencies. Applicationwise, we suggest a simple way to control the phase self-locking region by varying the intracavity third-order dispersion.