Spin transport in a magnetic tunnel junction with a synthetic antiferromagnetic (SAF) free layer is investigated using the drift-diffusion model. Although the diffusive transport is inappropriate for the MgO tunnel barrier, the drift-diffusion model is found to capture the core features of in-plane spin-transfer torque (STT) through the tunnel barrier, and more importantly, it can describe non-negligible STT exerting on two ferromagnets in a SAF-free layer. STT in a SAF-free layer substantially changes the magnetization dynamics and induces a shift of the critical switching current. STT in a SAF-free layer suppresses current-induced parallel-to-antiparallel switching, whereas it encourages antiparallel-to-parallel switching.