HTML isn't even actually context-free, though you can possibly parse it as context-free (type-2) with some kludges. The kludges are the same as the kludges for parsing it in a regular language, though! (I.e. "all" you need to do is add backreferences.)

Recognizing paired start/end tags is equivalent to recognizing palindromes, and the palindrome language is only context-free if there's a finite, fixed alphabet. But the set of HTML tags a parser needs to match isn't finite. While there are a finite number of defined tags in any particular version of the HTML specification, a parser is still supposed to be able to match start/end tags it doesn't recognize, to distinguish the case of malformed syntax from the case of correctly formed syntax that uses unrecognized tags.

An easy way to see it is to try to write out CFG productions for HTML. The intuitive way requires something like backreferences, which would make the language no longer context-free:

   html = '<' + tag + '>' + html + '</' + $0 + '>'

But in a context-free language you don't have that, so you instead have to do something like:

   html = '<head>' + html + '</head>'
        | '<body>' + html + '</body>'
        | '<b>' = html + '</b>'
        | ...

which of course only works if the number of tags is finite so that you can exhaustively enumerate them.