As you say, this is not a regex parser, but some readers may not know the theorem that the regular languages are exactly those which can be parsed by a Turing m

As you say, this is not a regex parser, but some readers may not know the theorem that the regular languages are exactly those which can be parsed by a Turing machine with constant memory, and parenthesis count can have log n bits. In fact, any Turing machine using o(log log n) space recognizes a regular language (so there is an equivalent machine using O(1) space).