Check the screenshot below:
This is very strange situation. Could you please try to deactivate and activate ( DO NOT uninstall the plugin, otherwise all the data will be lost) the Paid Membership plugin. If this does not help, you can PM the admin access details, so we could check into this issue for you.
Ilija, the option for moderators was added only in the latest version of the plugin. Before, the website admin only can change the membership.
Let me explain, the standard software behavior is when plugin installs - it adds section into the Moderators table with uncheked boxes for moderators. After the update, the plugin also can add the boxes, but unchecked according to the standard software behavior. To make the button appear for moderators, the website admin should check the boxes manually.
We have tested this at our test website once again and everything works with no issues.
Anyway, it is ok now, if it's settable via moderators.