In aerospace structures, parametric resonance or dynamic instability, a destructive phenomenon, can occur due to fluctuating excitation sources such as engine thrust. Auxetic materials and structures are increasingly developed in advanced industries such as aerospace and aviation. Due to their negative Poissons’s ratio, they show interesting static and dynamic behaviors. In this paper, an auxetic structure is used in the core of a thin sandwich cylindrical shell. The effect of geometric parameters of the unit cell of the cellular auxetic core structure is modeled according to the existing relation for the equivalent mechanical properties. Based on the classical shell theory, the dynamic stability of aluminum three layered sandwich cylindrical shells with auxetic core under combined static and periodic loading is investigated. Expansion of a normal mode for the equations of motion leads to a Mathieu-Hill system of equations. The Bolotin method is employed to determine the instability regions for solving the Mathieu-Hill equations. Validation of the natural frequency and dynamic stability results is performed by comparison with other researchers' publications. Then, parametric study is achieved by studying the influence of the geometric parameters of the unit cell on the dynamic stability of the shell. Results show that the corner angle and dimensional aspect ratio of the unit cell govern the size and origin frequency of the unstable areas.