Sometimes if you will click on Add a web part link then you may not see web part categories even if you have full control to the site where you want to add a web part.
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When you edit the page and try to add by clicking on the Add a web part link, you may not see the web part categories even if you have full control.
If you are working in SharePoint 2010, then you may see only lists and libraries. And if you are working in SharePoint 2013, then you may see only Apps category.
There might be a possibility is that, You are probably working in a sub site and do not have read access to the top level site in the site collection. You will need at least read access to the top level site's web part gallery.
If you grant READ rights to the Web Parts Gallery of the top level site to the sub site owners then all of the web parts in the gallery can be seen in the Add Web Part popup of the sub site.
So if due to some restrictions you will not be able to give access to the parent site, then grant rights to the Web Part Gallery only. After this when user will click on Add a web part, user will be able to see all the web part categories.