Superconducting coplanar waveguide (CPW) resonators are powerful and versatile tools used in areas ranging from radiation detection to circuit quantum electrodynamics. Their potential for low intrinsic losses makes them attractive as sensitive probes of electronic properties of bulk materials and thin films. Here we use superconducting MoRe CPW resonators to investigate the high-frequency (up to 0.3 GHz) and low-temperature (down to 3.5 K) permittivity of SrTiO3, a nonlinear dielectric on the verge of a ferroelectric transition (quantum paraelectricity). We perform a quantitative analysis of its dielectric properties as a function of external dc bias (up to ±15 V), rf power, and mode number and discuss our results within the framework of the most recent theoretical models. We also discuss the origin of a fatigue effect that reduces the tunability of the dielectric constant of SrTiO3, which we relate to the presence of oxygen vacancies.