In this work, we investigate numerically the propagation of Lamb waves in a film bulk acoustic resonator (FBAR) structure formed by piezoelectric ZnO layer sandwiched between two Mo electrodes coupled with Bragg reflectors; the system is thus considered as a phononic-crystal (PnC) plate. The aim is to suppress the first-order symmetric Lamb wave mode considered as a spurious mode caused by the establishment of a lateral standing wave due to the reflection at the embedded lateral extremities of the structure; this spurious mode is superposing to the main longitudinal mode resonance of the FBAR. The finite element study, using harmonic and eigen-frequency analyses, is performed on the section of FBAR structure coupled with the PnC. In the presence of PnC, the simulation results show the evidence of a selective band gap where the parasitic mode is prohibited. The quality factor of the FBAR is enhanced by the introduction of the PnC. Indeed, the resonance and antiresonance frequencies passed from 1000 and 980 (without PnC) to 2350 and 1230 (with PnC), respectively. This is accompanied by a decrease in the electromechanical coupling coefficient from 10.60% to 6.61%.