The maximum leaf spanning tree problem consists in finding a spanning tree of a graph that maximizes the number of leaves that the tree has. This problem has been found to be NP-hard for general graphs. It has several relevant applications in the context of telecommunication networks. In this paper, we tackle this problem by proposing the use of a parallel algorithm based on the strategic oscillation methodology. In particular, we propose two different parallel approaches and we compare our best variant with previous algorithms of the state of the art. The proposed approach outperforms previous ones in the state of the art, which is also confirmed by the use of statistical tests.