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We consider the problem of finding all minimal transversals of a hypergraph H@?2^V, given by an explicit list of its hyperedges. We give a new decomposition technique for solving the problem with the following advantages: (i) Global parallelism: for certain classes of hypergraphs, e.g., hypergraphs of bounded edge size, and any given integer k, the algorithm outputs k minimal transversals of H in time bounded by polylog(|V|,|H|,k) assuming poly(|V|,|H|,k) number of processors. Except for the case of graphs, none of the previously known algorithms for solving the same problem exhibit this feature. (ii) Using this technique, we also obtain new results on the complexity of generating minimal transversals for new classes of hypergraphs, namely hypergraphs of bounded dual-conformality, and hypergraphs in which every edge intersects every minimal transversal in a bounded number of vertices.