Scenario:
Open the page in edit mode by adding some query string parameters to the url
Simply append the following to the URL:
?ToolPaneView=2&pagemode=edit
Your url:
http://site:port/web/page.aspx
... will become:
http://site:port/web/page.aspx?ToolPaneView=2&pagemode=edit
The page will open in edit mode with the "Add Web Parts" web part visible.
I tried but it didn't working with Chrome or Opera. Why not?