This paper deals with a design problem of an output feedback control system with a parallel feedforward compensator (PFC) for discrete-time systems. A design scheme of an almost strictly positive real (ASPR) based output feedback control system with an adaptively adjusting PFC is proposed. In the proposed method, the parameters of a PFC are adaptively adjusted for remaining the ASPR-ness of the resulting augmented system in order to guarantee the stability of the designed control system. The method makes it possible to design a PFC without using some kind or another priori information about the controlled system. The stability of the obtained control system is analyzed theoretically and the effectiveness of the proposed method will be confirmed through numerical simulations.