The whole argument hinges on one word in your post: arbitrary.
I parse my own HTML I produce directly in a context where I fully control the output. It works fine, but parsing other people’s HTML is a lesson in humility. I’ve also done that, but I did it as a one time thing. I parsed a specific point in time, refusing to change that at any point.
It also hinges on another word: parsing. There are things other than parsing that you might want to do. For example, if you want to count the number of `<hr>` tags in an HTML document, that doesn't require parsing it, and can indeed be done with regex.