We describe an approach to LFG parsing that is optimized for c-structure discontinuities that are established through "zipper" unification. These are characterized by parallel c-structure paths that carry the same function assignments. Wedekind and Kaplan (2020) demonstrated that LFG grammars giving rise to discontinuities with finitely bounded zipper paths can express only mildly context-sensitive dependencies and thus can be converted to equivalent linear context-free rewriting systems (LCFRSs). In principle, parsing with LCFRS grammars can be accomplished in polynomial time, but that may not be the most effective way of parsing with mildly context-sensitive dependencies. In this paper we propose a hybrid strategy for LFG parsing that is tuned to the common case of bounded zippers but still allows for putatively rare constructions that do not conform to the formal restrictions that guarantee finite boundedness. This strategy automatically takes advantage of mildly context-sensitive dependencies in addition to the context-free dependencies that the XLE parsing system has focused on (Maxwell and Kaplan 1996).