This paper reports an analytical solution to one of the problems related to applied mechanics and acoustics, which tackles the analysis of free axisymmetric bending oscillations of a circular plate of variable thickness. A plate rigidly-fixed along the contour has been considered, whose thickness changes by parabola h(ρ)=H 0 (1+µρ) 2 . For the initial assessment of the effect exerted by coefficient μ on the results, the solutions at μ=0 and some μ≠0 have been investigated. The differential equation of the shapes of a variable-thickness plate's natural oscillations, set by the h(ρ) function, has been solved by a combination of factorization and symmetry methods. First, a problem on the oscillations of a rigidly-fixed plate of the constant thickness (μ=0), in which h(1)/h(0)=η=1, was solved. The result was the computed natural frequencies (numbers λ i at i=1...6), the constructed oscillation shapes, as well as the determined coordinates of the nodes and antinodes of oscillations. Next, a problem was considered about the oscillations of a variable-thickness plate at η=2, which corresponds to μ=0.4142. Owing to the symmetry method, an analytical solution and a frequency equation for η=2 were obtained when the contour is rigidly clamped. Similarly to η=1, the natural frequencies were calculated, the oscillation shapes were constructed, and the coordinates of nodes and antinodes of oscillations were determined. Mutual comparison of frequencies (numbers λ i ) shows that the natural frequencies at η=2 for i=1...6 increase significantly by (28...19.9) % compared to the case when η=1. The increase in frequencies is a consequence of the increase in the bending rigidity of the plate at η=2 because, in this case, the thickness in the center of both plates remains unchanged, and is equal to h=H 0 . The reported graphic dependences of oscillation shapes make it possible to compare visually patterns in the distribution of nodes and antinodes for cases when η=1 and η=2. Using the estimation formulae derived from known ratios enabled the construction of the normalized diagrams of the radial σ r and tangential σ θ normal stresses at η=1 and η=2. Mutual comparison of stresses based on the magnitude and distribution character has been performed. Specifically, there was noted a more favorable distribution of radial stresses at η=2 in terms of strength and an increase in technical resource