Parent icon on menu item level not shown?

no idea, as I can not login and see the problem ... please give me a test account

How can I send the login to you ?

www.joomlack.fr/en/contact

ok ,the problem is that you are using a "heading" menu type. You shall use a "separator instead", it will look as the other items
then you have setup some custom styling in the menu item params, so if you have created another menu item for the registered, you must also setup the styling as you want in this item params

The reason for choosing heading as menu item type was that as you now see, the orange color is not being set on the text, when using a separator menu item type ???

you can setup the color in the menu item > maximenu params as you already did
anyway, you must setup the color in the same way for hte logoff item