Capitalizing on the graph-theoretic account of logic systems and their fibrings in [19], and inspired on notions of concurrency, a novel form of combination of logics — parallel composition — is proposed at the semantic level. Parallel composition allows both symmetric (like sharing) and non-symmetric (like triggering) forms of interaction to be specified at the signature level. Parallel composition subsumes fibring as a special case. Parallel composition is presented using universal constructions in the categories of signatures and interpretations, using a cofibration for synchronization. The conservative nature of parallel composition under lossless synchronization is established using a generalized bisimulation technique. The conservative nature of free parallel composition and unconstrained fibring follow as corollaries. Joint work with Cristina Sernadas and João Rasga.