The performance of superconducting devices is often degraded by the uncontrolled appearance and disappearance of quasiparticles, a process known as poisoning. We demonstrate the electrostatic control of quasiparticle poisoning in the form of single-charge tunneling across a fixed barrier onto a Coulomb island in an InAs/Al hybrid nanowire. High-bandwidth charge sensing was used to monitor the charge occupancy of the island across Coulomb blockade peaks, where tunneling rates were maximal, and Coulomb valleys, where tunneling was absent. Electrostatic gates changed the on-peak tunneling rates by two orders of magnitude for a barrier with fixed normal-state resistance, which we attribute to the gate dependence of the size and softness of the induced superconducting gap on the island, corroborated by separate density-of-states measurements. Temperature and magnetic field dependence of tunneling rates are also investigated.