You can't parse [X]HTML with regex. Because HTML can't be parsed by regex. Regex is not a tool that can be used to correctly parse HTML.
You can parse a subset of it though, like if you're in control of the html yourself and avoid certain structures
Parsing HTML with a regex is never a good option, but it's sometimes the only option.
You can absolutely parse HTML with regex, so long as the document is finite in length. Every finite language is regular, hence can be parsed with regex's.
Tony the pony, he comes.
I do not count substitution as "parsing"
Some commenters are missing that this is a reference to https://stackoverflow.com/a/1732454.