Symptoms: A user with design or full control permission attempts to add a web part to a page, but finds that there are no web parts available for her to select. The dialog box provides only App Parts in the selection list.
Problem: The user does not have read permission to the top level site of the site collection. Read permission to the site collection is required because there is only one Web Part Gallery per site collection, and it is housed on the parent site.
Solution: Grant at least Read permission for the site collection to the user with the issue.