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The spin Hamiltonian parameters for the two Cu$^{2+}$ centers $A1$ and $A2$ in the calcined catalysts CuO-ZnO are theoretically investigated using the high order perturbation formulas of these parameters for a $3d^{9}$ ion in tetragonally elongated octahedra. In the above formulas, the tetragonal field parameters $Ds$ and $Dt$ are determined from the superposition model, by considering the relative axial elongation of the oxygen octahedron around the Cu$^{2+}$ due to the Jahn-Teller effect. Based on the calculations, the relative elongation ratios of about 5% and 3% are obtained for the tetragonal Cu$^{2+}$ centers $A1$ and $A2$, respectively. The theoretical spin Hamiltonian parameters are in good agreement with the observed values for both systems. The larger axial elongation in center $A1$ is ascribed to the more significant low symmetrical (tetragonal) distortion of the Jahn-Teller effect. The local structures characterized by the above axial elongations are discussed.