Items like https://www.wikidata.org/wiki/Q27697316 display a thumbnail on Wikidata. The size of these thumbnails seems to be static and rather small hardcoded to 150px it seems ( https://upload.wikimedia.org/wikipedia/commons/thumb/2/21/Piombo_cristo_cruz_prado.jpg/150px-Piombo_cristo_cruz_prado.jpg ). In my preferences at https://www.wikidata.org/wiki/Special:Preferences#mw-prefsection-rendering I can set thumbnail from 120px to 400px. This should be used so a user can change it.
The thumbnail user preference can be retrieved through https://www.wikidata.org/w/api.php?action=query&meta=userinfo&uiprop=options&output=json . It looks like it returns an integer from 0 (120px) to 7 (400px) in the field "thumbsize".
Time-boxed to 4h for investigation of how to by-pass parser-cache for those images.