SharePoint 2013 Web Part Issue

I have a SharePoint 2013 Server that a user has Full control of a site, Publishing Feature is enabled, However if they user tries to add "Summary Links" (A Publishing Web part) they get access denied. If I set them as a Site Collection Admin it works fine.

They may have the same issue across all Publishing web parts, however I do not know which ones are in fact Publishing web parts.

Any thoughts?

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bbresslinAuthor Commented:
The problem with that, is once the Site Collection administrator adds the web part.
Any full control user or Contribute user can no longer edit the page.
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