Rare events play a key role in many applications and numerous algorithms have been proposed for estimating the probability of a rare event. However, relatively little is known on how to quantify the sensitivity of the probability with respect to model parameters. In this paper, instead of the direct statistical estimation of rare event sensitivities, we develop novel and general uncertainty quantification and sensitivity bounds which are not tied to specific rare event simulation methods and which apply to families of rare events. Our method is based on a recently derived variational representation for the family of Rényi divergences in terms of risk sensitive functionals associated with the rare events under consideration. Based on the derived bounds, we propose new sensitivity indices for rare events and relate them to the moment generating function of the score function. The bounds scale in such a way that we additionally develop sensitivity indices for large deviation rate functions.