The buckling and parametric resonance characteristics of laminated composite spherical sandwich shell panels with viscoelastic material (VEM) core are investigated in the present analysis considering full geometric nonlinearity in the Green–Lagrange sense. The study includes the longitudinal strain and normal strain in the transverse direction along with transverse shear deformation of the VEM core. The core displacements are considered to be varying linearly along the thickness and those of the face sheets follow first-order shear deformation theory. An eight-noded sandwich shell finite element of the serendipity family is adopted to discretize the sandwich shell panel domain. The finite element-based equation of motion is derived using Hamilton’s principle in the form of the Mathieu–Hill-type equation. The dynamic instability regions are obtained by applying Hsu’s criteria-based Saito–Otomi conditions to the transformed equation motion. An in-house finite element-based code is developed in the MATLAB platform to solve the stability problem and to establish the stability regions. A parametric study is carried out to investigate the influence of different system parameters on the critical buckling load and the parametric resonance of the sandwich shell panels. It is noted that an increase in core and constraining layer thicknesses increases the critical buckling load of the sandwich shell panels. The stability boundaries are observed to shift toward a higher-excitation-frequency region in the stability diagram with an increase in constraining layer thickness and a decrease in aspect ratio. © 2020, Springer-Verlag GmbH Austria, part of Springer Nature.