We study the class of masking based domain extenders for UOWHFs. Our first contribution is to show that any correct masking based domain extender for UOWHF which invokes the compression UOWHF s times must use at least ⌈log2 s⌉ masks. As a consequence, we obtain the key expansion optimality of several known algorithms among the class of all masking based domain extending algorithms. Our second contribution is to present a new parallel domain extender for UOWHF. The new algorithm achieves asymptotically optimal speed-up over the sequential algorithm and the key expansion is almost everywhere optimal, i.e., it is optimal for almost all possible number of invocations of the compression UOWHF. Our algorithm compares favourably with all previously known masking based domain extending algorithms.