How do I determine the resonance frequency of this circuit
Regarding series versus parallel resonance: only when a complete path exists, for the circulating currents, can resonance occur. In parallel resonance, the L and C are obviously in a tight loop.
In series resonance, in some oscillators, the "circulating currents" take a path that included the VDD wiring, and the oscillator will fail **unless* a low-loss capacitor is used in VDD bypassing.