Electronic devices may be provided that include radio-frequency transceiver circuitry and antennas. An antenna may be formed from an antenna resonating element and an antenna ground. The antenna resonating element may have a shorter portion that resonates at higher communications band frequencies and a longer portion that resonates at lower communications band frequencies. The resonating element may be formed from a peripheral conductive electronic device housing structure that is separated from the antenna ground by an opening. A parasitic monopole antenna resonating element or parasitic loop antenna resonating element may be located in the opening. Antenna tuning in the higher communications band may be implemented using an adjustable inductor in the parasitic element. Antenna tuning in the lower communications band may be implemented using an adjustable inductor that couples the antenna resonating element to the antenna ground.