I am in Department A that has a Parent site and multiple subsites
There is also a Department B that has its own Parent site and multiple subsites. I have read access to Department B's site. I am owner of Department A site.
Department B would like me to add their calendar from their parent site into my site (Department A) as a web part. Is this possible? If so, how?
Site collections (what you call "parent sites") are partionned containers, and can hardly communicate data accros.
With dev (creating some kind of Web part), we could achieve what you describe, but that would really be against the SharePoint "phylosophy"...
However, here's a simple "hack" you can easily do:
Navigate to the full calendar view on site B
Get the complete URL of the calendar view from the address bar
Add "?IsDlg=1" to the URL (this will prevent the menus/headers... all the chrome from appearing later in the iFrame we'll create at steps #4/#5) and copy the URL
If you are using SP 2013 you do have the option to do calendar overlays, which will allow you to plug an offsite calendar into an existing site calendar and overlay it. I've also done this in SP 2007 using a site feature (found on codeplex, i believe).