We show that the notoriously difficult problem of finding the minimum number of paths that cover the vertices of a graph can be solved efficiently for cographs. Our result implies that for this class of graphs finding a hamiltonian path and a hamiltonian cycle can be solved efficiently in parallel. Specifically, with an n-vertcx cograph G represented by its parse tree as input, our algorithm determines the number of paths in a minimum path cover in O(log n) time using n/log nprocessors in the EREW-PRAM; we also exhibit all the paths in a minimum path cover of G in O(log²n) lime using n/logn processors in the EREW-PRAM. Our result significantly improves on the state of the art.