This has been proposed as CGI task https://www.google-melange.com/gci/task/view/google/gci2014/5857599308169216

The parsed version of a wiki page can be obtained using the parse module
https://en.wikipedia.org/w/api.php?action=parse&page=Benjamin_Franklin

Copying the GCI task description:

This task is to create a new script to harvest data from HTML microformats in Wikipedia pages and other webpages, and add the data to items in Wikidata. The new script will be similar in nature to the existing script harvest_template.py, except it will use HTML instead of wikitext, and it can offer automatic assignments of values to properties where the microformats describe the data in a standardised way that maps to properties on Wikidata.

In T78416#938277, @murfel wrote:

I think to implement it in the following way: catch all page which link to a given template, get HTML for each page, look for table with id="template_name" inside of HTML, parse key-values in the table and add them to Wikibase.

Did I get it right?

maybe, but maybe not. My inclusion of {{Persondata}} as an example was perhaps misleading.

this harvest_microformats script should not be based on templates, as is the job of harvest_template.py .

This script will use pagegenerators as arguments to select which pages should be processed, and -page:"..." is the easiest to use for testing.

For each page, get the HTML as you've said, and look for microformats (http://microformats.org/) in the HTML. Microformats are usually described using HTML class:".." attributes, such as:

view-source:https://en.wikipedia.org/wiki/Benjamin_Franklin

<span class="bday">1706-01-17</span>
<span class="dday deathdate">1790-04-17</span>

and

view-source:https://en.wikipedia.org/wiki/Manchester_Ship_Canal

<th colspan="2" class="fn org" style="text-align:center;font-size:125%;font-weight:bold;font-size: larger; background-color: #CEDEFF">Manchester Ship Canal</th>

The two most important standardised microformats are
http://microformats.org/wiki/hCard
http://microformats.org/wiki/hCalendar

Another icroformat that is very relevant to wikis is http://microformats.org/wiki/rel-license

However Wikimedia mostly uses its own non-standard microformats, for example, "licensetpl" is used by Wikisource and Wikimedia Commons instead of rel-license

view-source:https://en.wikisource.org/wiki/The_Clipper_Ship_Era

<table class="licensetpl" style="display:none;">
<tr>
<td><span class="licensetpl_short">Public domain</span><span class="licensetpl_long">Public domain</span><span class="licensetpl_link_req">false</span><span class="licensetpl_attr_req">false</span></td>
</tr>
</table>

When microformats have been found in the HTML, yes .... "parse key-values [from the microformat] and add them to Wikibase" , but .. there are python libraries that already do most of the grunt work for you, so hopefully you dont need to do the parsing yourself, e.g. see http://microformats.org/wiki/parsers and search https://pypi.python.org/pypi/ . One library mentioned is https://github.com/tommorris/mf2py , which is maintained by @tommorris , English Wikipedia admin among other things.

A useful tool to see microformats in any webpage (courtesy of @murfel on IRC)
https://mf2py.herokuapp.com/parse?url=https://en.wikipedia.org/wiki/Benjamin_Franklin

The version packaged on pypi (0.2.1) doesnt handle wikipedia microformats. You will need to install the latest from github : https://github.com/tommorris/mf2py