We propose a parallel algorithm for local, on the fly, model checking of a fragment of CTL that is well-suited for modern, multi-core architectures. This model-checking algorithm takes benefit from a parallel state space construction algorithm, which we described in a previous work, and shares the same basic set of principles: there are no assumptions on the models that can be analyzed; no restrictions on the way states are distributed; and no restrictions on the way work is shared among processors. We evaluate the performance of different versions of our algorithm and compare our results with those obtained using other parallel model checking tools. One of the most novel contributions of this work is to study a space-efficient variant for CTL model-checking that does not require to store the whole transition graph but that operates, instead, on a reverse spanning tree.