In this correspondence, we investigate the joint optimization for an uplink reconfigurable intelligent surface (RIS)-aided millimeter wave integrated access and backhaul network (mmWave IABN) to maximize the spectral efficiency (SE). When the bandwidth partition ratio between the access part and the backhaul part is given, the sum end-to-end rate maximized problem is firstly formulated and equivalently transformed as the SE maximization oriented joint optimization under the constraints including the users' transmission power and signal-to-interference-plus-noise-ratio (SINR) requirement, and the RIS's unit mode. Secondly, an alternating iterative optimization approach is designed to obtain a nearly-optimal solution by solving the two subproblems (i.e., power control at the users and phase-shift manipulation at the RIS), which are decomposed from the original problem based on the successive convex approximation. Thirdly, the sum SE superiority of the proposed approach is demonstrated through the comparison with other baselines under different settings. In addition, the convergence of the proposed approach and the impact of the bandwidth partition ratio on the system performance are validated.