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In this paper, we study an HP TiO₂ memristor model with a parasitic memcapacitor, and it is shown that the parasitic element has significant effects on the volt-ampere characteristics and the dynamics of the memristor circuits. Further, the study shows that the pinched point on the hysteresis loop of the HP TiO₂ memristor may deviate from the origin. We also show that the memristor with a parasitic memcapacitor will eventually lose its stored flux when its power is switched off. Furthermore, in order to study the parasitic effects in circuits, we derive a simple three-element series circuit that contains an inductor, a negative resistor, and an HP TiO₂ memristor. If the parasitic memcapacitor of the memristor is considered, we observe a parasitic periodic oscillation occurred in the circuit.