This is a result of a slight change in the HTML structure of headings. Previously, they looked like this:

<h2>
    <span class="mw-headline" id="Italian">Italian</span>
    <span class="mw-editsection">...</span>
</h2>

whereas Parsoid renders them as:

<div class="mw-heading mw-heading2" id="mwEA">
    <h2 id="Italian">Italian</h2>
    <span class="mw-editsection" id="mwEQ">...</span>
</div>

The CSS selectors that the gadget uses (at https://en.wiktionary.org/wiki/MediaWiki:Gadget-WiktCountryFlags.css) look like .ns-0 h2 #Italian::before, and so match the old-style headings but not the new ones.

This task may be worthwhile as "this is the sort of gadget that heading style changes will break", but the best fix would be to adjust the gadget by adding other selectors, so .ns-0 h2 #Italian::before becomes .ns-0 h2 #Italian::before, .ns-0 h2#Italian::before, which works on both Parsoid and non-Parsoid HTML. I updated the script that generates the gadget source; all that's needed now is for an enwiktionary intadmin to re-run the generating script.