Mode-coupled vibrations in an ultra-high frequency (UHF) ZnO thin film bulk acoustic resonator (FBAR) operating at thickness-extensional (TE) mode are studied by employing weak boundary conditions (WBCs), constructed based on Saint-Venant's principle and mixed variational principle in the piezoelectric theory. The frequency spectra, describing the lateral size-dependence of mode couplings between the main mode (TE) and undesirable eigen-modes, for clamped lateral edges are compared with the existing frequency spectra for free lateral edges to illustrate the boundary influence. The displacement and stress variations in FBAR volume are also presented to intuitionally understand and distinguish the difference of frequency spectra between these two different lateral edges, and then we discuss how to select outstanding lateral sizes to weaken the mounting effect. The frequency spectra predicted from our approximate WBCs are also compared with and agree well with those predicted by the finite element method (FEM) using COMSOL, which proves the correctness and accuracy of our theoretical method. These results indicate that the WBCs could have potentials in the valid predictions of lateral size-dependence of mode couplings in piezoelectric acoustic wave devices.