Isotropic viscoelastic plates of variable thickness under the effect of a uniformly distributed vibration load applied along one of the parallel sides, resulting in parametric resonance (with certain combinations of eigenfrequencies of vibration and excitation forces) are considered in the paper. It is believed that under the effect of this load, the plates undergo the displacements (in particular, deflections) commensurate with their thickness. Geometrically nonlinear mathematical model of the problem of parametric oscillations of a viscoelastic isotropic plate of variable thickness is developed using the classical Kirchhoff-Love hypothesis. Corresponding nonlinear equations of vibration motion of plates under consideration are derived (in displacements). The technique of the nonlinear problem solution by applying the Bubnov-Galerkin method at polynomial approximation of displacements (and deflection) and a numerical method that uses quadrature formula are proposed. The Koltunov-Rzhanitsyn kernel with three different rheological parameters is chosen as a weakly singular kernel. Parametric oscillations of viscoelastic plates of variable thickness under the effect of an external load are investigated. The effect on the domain of dynamic instability of geometric nonlinearity, viscoelastic properties of material, as well as other physical-mechanical and geometric parameters and factors (initial imperfections of the shape, aspect ratios, thickness, boundary conditions, excitation coefficient, rheological parameters) are taken into account. The results obtained are in good agreement with the results and data of other authors. The convergence of the Bubnov-Galerkin method is verified.