Although much has been said about parallelism in discourse, a formal, computational theory of parallelism structure is still outstanding. In this paper, we present a theory which given two parallel utterances predicts which are the parallel elements. The theory consists of a sorted, higher-order abductive calculus and we show that it reconciles the insights of discourse theories of parallelism with those of Higher-Order Unification approaches to discourse semantics, thereby providing a natural framework in which to capture the effect of parallelism on discourse semantics.