Electronic devices are provided that contain wireless communications circuitry. The wireless communications circuitry may include radio-frequency transceiver circuitry and antenna structures. A parallel-fed loop antenna may be formed from portions of a conductive bezel and a ground plane. The antenna may operate in multiple communications bands. The bezel may surround a peripheral portion of a display that is mounted to the front of an electronic device. The bezel may contain a gap. Antenna feed terminals for the antenna may be located on opposing sides of the gap. A variable capacitor may bridge the gap. An inductive element may bridge the gap and the antenna feed terminals. A switchable inductor may be coupled in parallel with the inductive element. Tunable matching circuitry may be coupled between one of the antenna feed terminals and a conductor in a coaxial cable connecting the transceiver circuitry to the antenna.