Series and parallel resonance are two distinct circuit structures. In a parallel circuit, there is infinite resistance to an alternating current. In a series circuit, there is zero resistance to an alternating current, but only when the greatest possible response to a varying signal occurs. Resonance itself is the maximization or minimization of an electrical current at a particular frequency.
Bpa.gov explains the conditions for both series and parallel resonance. In series resonance, if a circuit's capacitance (the ability to store an electric charge) and inductance (the change in current flow by electromotive forces) both pass through the components of a device sequentially, the device to which the circuit belongs displays low resistance to an alternating current at a level where the greatest response is produced by the circuit. The current flow through the device is maximized at this point. Parallel resonance is more involved than circuit resonance. In parallel resonance, if the circuit capacitance and inductance connect to a common point at each end of the circuit, current flow is reduced to a certain frequency. Currents of a larger magnitude move continuously within the capacitor, which is a device used to store electrical charges, and the inductor, which is a component of the device that creates electromotive forces by varying the current flow throughout the device.