An electronic device may have a housing in which an antenna is mounted. An antenna window may be mounted in the housing to allow radio-frequency signals to be transmitted from the antenna and to allow the antenna to receive radio-frequency signals. Near-field radiation limits may be satisfied by reducing transmit power when an external object is detected in the vicinity of the dielectric antenna window and the antenna. A capacitive proximity sensor may be used in detecting external objects in the vicinity of the antenna. The proximity sensor may have conductive layers separated by a dielectric. A capacitance-to-digital converter may be coupled to the proximity sensor by inductors. The capacitive proximity sensor may be interposed between an antenna resonating element and the antenna window. The capacitive proximity sensor may serve as a parasitic antenna resonating element and may be coupled to the housing by a capacitor.