In parsing phrase structures, supertagging achieves a symbiosis between the interpretability of formal grammars and the accuracy and speed of more recent neural models.The approach was only recently transferred to parsing discontinuous constituency structures with linear context-free rewriting systems (LCFRS).We reformulate and parameterize the previously fixed extraction process for LCFRS supertags with the aim to improve the overall parsing quality.These parameters are set in the context of several steps in the extraction process and are used to control the granularity of extracted grammar rules as well as the association of lexical symbols with each supertag.We evaluate the influence of the parameters on the sets of extracted supertags and the parsing quality using three treebanks in the English and German language, and we compare the best-performing configurations to recent state-of-the-art parsers in the area.Our results show that some of our configurations and the slightly modified parsing process improve the quality and speed of parsing with our supertags over the previous approach.Moreover, we achieve parsing scores that either surpass or are among the state-of-the-art in discontinuous constituent parsing.