The authors of the HTML indentation script shipped by Vim seem to have decided not to indent inside tags such as
<p> because they are "self-closing" tags, meaning that the closing
</p> is optional. I imagine the rationale for that is that, if a
</p> was omitted by the user, the indentation wouldn't have been retracted, so each new paragraph would get indented further and further.
See this comment in
Self-closing tags and tags that are sometimes self-closing (e.g.,
<p>) are not here (when encountering
</p> we can find the matching
<p>, but not the other way around).
Known self-closing tags:
You can override this behavior and extend the list of tags that increment indentation by setting the
g:html_indent_inctags to a comma-separated list of additional tags that should require indentation.
In your case:
let g:html_indent_inctags = 'p'