The "narrowest" generalization of Dyck languages that I am aware of is Visibly Pushdown languages. Are there any useful classes of languages that are intermediate between Dyck languages and Visibly Pushdown Automata?
For example, Dyck languages require parenthesis to be paired, but visibly pushdown languages merely require that stack movement is controlled by the input tape, without any requirement that "call symbols" and "return symbols" are matched one-to-one.
It is easy to imagine an intermediate class, e.g. the class of visibly push down languages where the removal of all level symbols from each word results in a Dyck language, but I'm not sure if any such classes have useful properties or have been the subject of previous study.