Multiword expressions (MWEs) are pervasive in natural languages and often have both idiomatic and compositional readings, which leads to high syntactic ambiguity. We show that for some MWE types idiomatic readings are usually the correct ones. We propose a heuristic for an A* parser for Tree Adjoining Grammars which benefits from this knowledge by promoting MWE-oriented analyses. This strategy leads to a substantial reduction in the parsing search space in case of true positive MWE occurrences, while avoiding parsing failures in case of false positives.