In transition-metal perovskites (ABO_{3}) most physical properties are tunable by structural parameters such as the rotation of the BO_{6} octahedra. Examples include the Néel temperature of orthoferrites, the conductivity of mixed-valence manganites, or the band gap of rare-earth scandates. Since oxides often hold large internal electric dipoles and can accommodate heavy elements, they also emerge as prime candidates to display Rashba spin-orbit coupling, through which charge and spin currents may be efficiently interconverted. However, despite a few experimental reports in SrTiO_{3}-based interface systems, the Rashba interaction has been little studied in these materials, and its interplay with structural distortions remains unknown. In this Letter, we identify a bismuth-based perovskite with a large, electrically switchable Rashba interaction whose amplitude can be controlled by both the ferroelectric polarization and the breathing mode of oxygen octahedra. This particular structural parameter arises from the strongly covalent nature of the Bi-O bonds, reminiscent of the situation in perovskite nickelates. Our results not only provide novel strategies to craft agile spin-charge converters but also highlight the relevance of covalence as a powerful handle to design emerging properties in complex oxides.