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An ear-decomposition of a digraph is a representation of it as the union of (open or closed) directed paths, each having its endpoints in common with the union of the previous paths but nothing else. We prove that finding an ear-decomposition of a strongly directed graph is in NC, i.e. an eardecomposition can be constructed in parallel in polylog time, using a polynomial number of processors. Using a similar technique, we show that the problem of finding a minimum weight spanning arborescence in an arcweighted rooted digraph is in NC.