This paper introduces a novel multi-user mobile edge computing (MEC) scheme facilitated by the simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) and the unmanned aerial vehicle (UAV). Unlike existing MEC approaches, the proposed scheme enables bidirectional offloading, allowing users to concurrently offload tasks to the MEC servers located at the ground base station (BS) and UAV with STAR-RIS support. Specifically, we formulate an optimization problem aiming at maximizing the energy efficiency of the system while ensuring the quality of service (QoS) constraints by jointly optimizing the resource allocation, user scheduling, passive beamforming of the STAR-RIS, and the UAV trajectory. A block coordinate descent (BCD) iterative algorithm designed with the Dinkelbach's algorithm and the successive convex approximation (SCA) technique is proposed to effectively handle the formulated non-convex optimization problem with significant coupling among variables. Simulation results indicate that the proposed STAR-RIS enhanced UAV-enabled MEC scheme possesses significant advantages in enhancing the system energy efficiency over other baseline schemes including the conventional RIS-aided scheme.