Send As permission on Office 365 not with Add-ADPermission
With an On-Premises Exchange 2010 environment an administrator hands out Send-As permission with the Add-ADPermission cmdlet.
But I found out that this doesn’t work in Office 365. In this case a P plan (no Dirsync, Federation or such), you can see the result in the screenshot:
For handing out Send-As permissions (and other rights) in Office 365, you’ll have to use Add-RecipientPermission. And cmdlet does not work in a On-Premise Exchange environment
Also see: Office 365 Help Give Users Send As Permission