A large part of wide coverage Tree Adjoining Grammars (TAG) is formed by trees that satisfy  the restrictions imposed by Tree Insertion Grammars (TIG). This characteristic can be used to  reduce the practical complexity of TAG parsing, applying the standard adjunction operation only  in those cases in which the simpler cubic-time TIG adjunction cannot be applied. In this paper,  we describe a parsing algorithm managing simultaneous adjunctions in TAG and TIG