The first option is easy to rule out: HTML is not a regular language. Evidence: S-Expressions are not a regular language. XML is isomorphic to S-expressions of the form (<tagname> ((<attr1> <value1>) (<attr2> <value2>) ...) <contents>). HTML is isomorphic to XHTML, which is XML.[1]
I can't (read: don't feel like trying to) rigorously disprove that Perl's "regular expression" facility could be an HTML parser all by itself, but it looks like what's going on here is closer to a regex-based tokenizer (certainly possible, and in fact this is a very common pattern). Then, regular Perl flow control constructs are used to interpret the tokens as tags and such.
[1] Edit: Looking at that, it's not as solid as it was in my head when I was writing it. HTML is still not a regular language, because regular languages cannot key on nesting with uncapped depth.
Was he able write an arbitrary HTML parser using regexes because HTML is a regular language or because Perl's string matching syntax, which can match some languages that are not regular, is called "regular expressions" anyway?
I am pretty sure that, for example, an arbitrary number of balanced "<div></div>" pairs are not a regular expression however you can use modern PCRE to match it anyway.