In this paper, we study variants of the parallel hash function construction of Damgård. We first show an improvement such that the number of processors is almost a half if |M|=(2s + 1)n for some s, where M is the message to be hashed. We next show that there exists a variant of our parallel hash construction such that it is secure even if the underlying compression function is not necessarily collision-free nor one-way. The cost is that some constant times more processors are required.