Are all finitely recursive context free languages parseable with a regexp? Let's say I have a context free language. It can be recognised by a pushdown automaton. Chances are it can't be parsed with a regular expression, as regular expressions are not as powerful as pushdown automata.

Now, let's put an additional constraint on the language: the maximum recursion amount must be finite.

Because the stack size has an upper bound in this case, my understanding is that there are finite number of stack configurations reachable. This means I could number them 0, 1, 2, 3, ..., N. So, I should be able to create a deterministic finite automaton (DFA) with states 0, 1, 2, 3, ..., N that recognises the same language that the pushdown automaton recognises.

Now, if I'm able to create an equivalent DFA, doesn't it mean that there exists a regular expression that can parse the context-free language with maximum recursion amount?

So, my theory is that all context-free languages that have a maximum recursion amount can be parsed with regular expressions. Is this theory correct? Of course, the theory says nothing about the complexity of the regular expression, it just says such a regular expression should exist.

So, in other words: if your stack memory is limited, a regexp can do the job of a HTML/XML parser!

In principle, isn't it true that computers with finite memory are actually DFAs and not Turning machines?
We can take it even further: if we put a limit on the size of the HTML/XML, say 1PB, then there is only a finite number of them, so we can trivially parse them in $O(1)$ using a giant look-up table. However, that doesn't seem to tell us much about the complexity of parsing HTML/XML in practice. 

The issue at stake here is modeling. A good model abstracts away the salient points of a real world situation in a way that makes the situation amenable to mathematical inquiry. Modeling HTML/XMLs as instances of arbitrarily recursing language forms a better model in practice than your suggestion or my suggestion.