What would you get if you add parameters to context free grammars?
Affix grammars (parameterised context-free grammars) were studied extensively by the eminent Dutch computer scientist Cornelis HA Koster, starting with his 1962 paper "Basic English, a generative grammar for a part of English", co-written with LGLT Meertens. In 1970, he produced a formalism of the concept; a useful overview is available in his 1971 paper "Affix Grammars for Programming Languages", a version of which I found on Citeseer.

In that paper, Koster compares his formalism (and another similar one) with Van Wijngaarden two-level grammars, and finds them to be very similar.

Dick Grune's invaluable annotated bibliography of parsing techniques includes a large number of other useful references for affix grammars and other non-Chomskyian formalisms. (See section 18.2.6 of the bibliography, although there are useful papers in other sections.) Grune covers affix grammars briefly in &sect;15.3.2 of the second edition of Parsing Techniques: A Practical Guide (and even more briefly in the first edition, available online) mentioning the fact that it is easy to adapt top-down (and other) parsing techniques.

Unless the domain of parameters is finite, in which case they can be removed by statically generating all possible productions resulting in a CFG, the class of affix grammars is certainly a strict superset of the set of context-free grammars. (An affix grammar for $a^n b^n c^n$ can be found in Dick Grune's Parsing Techniques reference above.)

Koster, who was also an editor of the Algol 68 report, was the original developer of the Compiler Description Language (CDL), based on his ideas about affix grammars. This toolkit and its later derivatives were used in production for many years. This page, which I found with a Google search and whose permanence I cannot guarantee, has links to the manual and download site for CDL3.