Analysis and numerical results for the axisymmetric vibrations of annular sandwich plates with relatively stiff core of non-uniform thickness have been investigated using a refined theory. The face sheets are treated as membranes of constant thickness and the core is assumed to be solid as well as moderately thick of parabolically varying thickness. Due to this-type of variation in the thickness of the core, the face sheets take the shape of paraboloid of revolution and because of this, the face sheets membrane forces contribute to the bending and transverse shear stresses of the core of the plate. The equations of motion for such a plate have been derived using Hamilton's energy principle. The frequency equations for three different combinations of boundary conditions; clamped at the inner edge and clamped or simply supported or free at the outer edge are obtained employing differential quadrature method. The lowest three roots of these frequency equations have been reported as the frequencies for the first three modes of vibration. The effect of various plate parameters such as radii ratio, taper parameter, thickness of the facings and core at the centre on the natural frequencies has been studied. Three-dimensional mode shapes for the specified plates have been illustrated. A comparison of the results with published work has been made. © 2015 Elsevier Ltd. All rights reserved.