Yes, that's what I mean by "recursive generations". L-System strings are interpreted in successive generations where the output of generation n becomes the input of generation n+1. The recursion I refer to here is at the level of generation steps.

You can simulate this with a parser for e.g. CFGs like you say, but then you need an outer loop that feeds the parsed strings back to the parser.

There's also a subtlety that I omitted in my earlier comment- apologies! but it's really a bit subtle. In L-Systems, strings are made up of "variables" and "constants", which are closely related to nonterminals and terminals, but are not exactly the same: in phrase-structure grammars, non-terminal symbols cannot appear in strings, only in grammar rules.

So for instance, the Dragon Curve rule I give above, "f -> f+g", means "wherever 'f' appears in a string, replace it with 'f+g'". To get the same result in a phrase-structure grammar you need to further define "f" as a pre-terminal, expanding to a single "f" character.

So for a CFG-parser-with-an-outer-loop to work on an L-System one would also need to modify the L-System to be a full CFG. To make it clear, here's the definition of the Dragon Curve L-System, from my example above:

  Axiom: f
  Constants: +,-
  f -> f+g
  g -> f-g

And here is its redefinition as a CFG, with nonterminals capitalised:

  F -> F+G
  G -> F-G
  F -> f
  G -> g

I think it's easier to see that these are different objects. I bet we'd find that, for every L-System, there's a phrase-structure grammar that accepts the same strings (but not necessarily generates the same strings only) and that is a simple re-write of the L-System with variables replaced by pre-terminals, as I do above, kind of like a standard form (not normal; because not fully equivalent). That may even be something well-known by L-Systems folks, I don't know.

Btw, the Dragon Curve L-system is described as an OL-System, that matches a Regular language. The grammar above is context-free but that's OK, a CFG can parse strings of a Regular language. I believe L-System languages go up to context-free expressivity, but I'm not sure, there are also parameterised L-Systems that may go beyond that.