In this paper, a non-intrusive reduced-order model (ROM) for parametric reactor kinetics simulations is presented. Time-dependent ROMs are notoriously data intensive and difficult to implement when nonlinear multiphysics phenomena are considered. These challenges are exacerbated when parametric dependencies are included. The proper orthogonal decomposition mode coefficient interpolation (POD-MCI) ROM presented in this work can be constructed directly from lower-dimensional full-order model (FOM) outputs and is independent of the underlying model. This greatly alleviates the data requirement of many existing ROMs and can be used without modification on arbitrarily complex models or experimental data. The POD-MCI ROM is demonstrated on a number of examples and yields accurate characterizations of the parametric behaviors of both FOM outputs and derived quantities of interest within the selected parameter spaces, at extremely attractive computational speedup factors relative to FOMs.