An electronic device may be provided with wireless circuitry. The wireless circuitry may include one or more antenna structures and transceiver circuitry such as millimeter wave transceiver circuitry. Antenna structures in the wireless circuitry may include patch antennas that are organized in a phased antenna array. Each patch antenna may include an antenna resonating element and a parasitic element. The parasitic element for the patch antenna may have dielectric-filled openings formed between coplanar parasitic conductors. The parasitic conductors may include a central parasitic conductor, four rectangular parasitic conductors formed around the central parasitic conductor, and corner parasitic conductors formed at the corners of the parasitic element. The corner parasitic conductors may be non-rectangular. For example, the corner parasitic conductors may have first and second perpendicular edges and a straight or curved third edge that joins the first and second edges.