You have to add another field to the page type, and use it in the visible condition of your Web Part.
Hit the arrow icon to open the macro editor:
![Image Text](https://content.screencast.com/users/DavidTK/folders/Jing/media/02528f29-7075-4417-a912-eccc41fa710b/2019-10-07_1108.png)
And add the field you added on your page type; Something like:
{%CurrentDocument.GetValue("DocumentShowForm")|(identity)GlobalAdministrator%}
Note that if you change something on a Template ALL pages using that template will reflect this update.