We study the stability of thin walled circular cylindrical shells subject to radial pressure loading. Towards this end, we first develop the equations of motion of these shells with predominant radial deflections. We identify additional terms in the shell equations of motion which have mostly been neglected in existing studies and ascertain their importance in predicting the correct static buckling pressure. For finite cylinders, incorporation of these terms results in the best prediction of the static buckling pressure. Furthermore, we also find that the assumed relationships between the in-plane displacements of a generic point and the in-plane displacements of the mid-surface, which change from one shell theory to the other, play a critical role in determining the correct onset of the buckling instability under radial loading. In particular, the Donnell shell theory which is very popular for the study of dynamic problems related to cylindrical shells does not predict the correct buckling pressure. We find that the Fl{\"{u}}gge-Lur{\'{e}}-Byrne theory is best suited for this purpose. Accordingly, the shell equations with the additional terms and the in-plane displacements related as per the Fl{\"{u}}gge-Lur{\'{e}}-Byrne shell theory have been used for the study of parametric instability of a cylinder subject to a uniform radially fluctuating pressure. Stability charts with respect to different combinations of the forcing parameters viz. the static component of the pressure, the amplitude and frequency of the fluctuating component of the pressure have been presented which can serve as design guideline for shells subject to fluctuating radial loads.