In this paper, we develop a generic controlled alternate quantum-walk model (called CQWMP) by combining parity-dependent quantum walks with distinct arbitrary memory lengths and propose a hash function (called QHFM-P) based on this model. The statistical properties of the proposed scheme are stable with respect to the coin parameters of the underlying controlled quantum walks; with certain parameter values, the collision resistance property of QHFM-P is better than that of the state-of-the-art hash functions based on discrete quantum walks. Moreover, the proposed hash function can also maintain near-ideal statistical performance when the input message is of small length. Additionally, we derive a type of inappropriate initial states of hash functions based on one-dimensional one-particle quantum walks (with ordinary shift operator) on cycles, with which all messages will be mapped to the same hash value, regardless of the angles adopted by the coin parameters.