While arbitrary HTML with only a regex is impossible, it's sometimes appropriate to use them for parsing a limited, known set of HTML.
If you have a small set of HTML pages that you want to scrape data from and then stuff into a database, regexes might work fine. For example, I recently wanted to get the names, parties, and districts of Australian federal Representatives, which I got off of the Parliament's web site. This was a limited, one-time job.
Regexes worked just fine for me, and were very fast to set up.