The inverted-F antenna with a parasitic branch (IFA&PB) is a commonly used antenna design in contemporary metal-bezel smartphones. In this structure, the primary branch of the IFA is typically directly excited through the feed, while the parasitic branch is activated via electromagnetic coupling. However, the traditional single feed excitation method for the IFA&PB often results in the radiation efficiency local minimum, significantly impacting the antenna's overall performance. To address this issue, this paper proposes a novel parallel feed approach with two different feeding phases for the IFA&PB in smartphones. This innovative method aims to eliminate the radiation efficiency local minimum observed in conventional IFA&PB structures. To validate this method, a high efficiency MHB antenna, consisting of the IFA&PB and a parallel feed with two different feeding phases, is achieved without the radiation efficiency local minimum in the MHB frequency band (1.71-2.69 GHz). Furthermore, this design is extended by integrating a traditional LB antenna, covering the LB frequency band (0.704-0.96 GHz), resulting in an octa-band mobile antenna with high efficiency. A prototype of the proposed design demonstrated measured-6 dB impedance bandwidths of 270 MHz (0.69-0.96 GHz) and 1750 MHz (1.35-3.10 GHz), effectively covering the LB and MHB frequency bands. The prototype's measured average efficiencies were-7.7 dB (17%) and-4.3 dB (37%), respectively. The proposed parallel feed with two different feeding phases can eliminate the radiation efficiency local minimum of the IFA&PB structure, presenting a promising solution for modern metal-bezel mobile antennas.