I've just checked again, and my non-mod account also gets the permission denied message when trying to access the Private forums from the drop down forum list at the top of the page (in the blue ribbon) as per your screen shot.
HOWEVER, if you simply scroll to the bottom of the PPRuNe home page, past all the other forums, just after PPRuNe Social you should get to the Airline Specific Private Forums, and my non-mod account can see the locked forums and access the links to apply (you may need to click the arrow on the right to open the list). Please can you try that?
Thanks,
SD