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We describe a constraint-based automated planner named Transition Constraints for Parallel Planning (TCPP). TCPP constructs its constraint model from a redefined version of the domain transition graphs (DTG) of a given planning problem. TCPP encodes state transitions in the redefined DTGs by using table constraints with cells containing don't cares or wild cards. TCPP uses Minion the constraint solver to solve the constraint model and returns a parallel plan. We empirically compare TCPP with the other state-of-the-art constraint-based parallel planner PaP2. PaP2 encodes action successions in the finite state automata (FSA) as table constraints with cells containing sets of values. PaP2 uses SICStus Prolog as its constraint solver. We also improve PaP2 by using dont cares and mutex constraints. Our experiments on a number of standard classical planning benchmark domains demonstrate TCPP's efficiency over the original PaP2 running on SICStus Prolog and our reconstructed and enhanced versions of PaP2 running on Minion.