Currently it seems the inbox will only show one parent up for any task. We use Asana for some complicated projects and often have 3-4 levels of sub-tasks. For example…
Acme Website Project >> Development >> CMS Installation and Config >> Install Core CMS might be a single example task. All the inbox will show you is “CMS Installation and Config >> Install Core CMS” which means you have to click on the item to know what client this is for.