Starting from a permutation of {0,...,n-1} we compute in parallel with a workload of O(n log n) a compact data structure of size O(n log n). This data structure allows to obtain the associated permutation graph and the transitive closure and reduction of the associated order of dimension 2 efficiently. The parallel algorithms obtained have a workload of O(n + m log n) where m is the number of edges of the permutation graph. They run in time O(log^2 n) on a CREW PRAM