No, first party is you, second party is the person you're talking to. They're a man in the middle. They have access to your contacts, your emails, everything--they're a third party in the middle of your comms.
OK so I guess since I use third party email servers this question doesn't really apply to me. I will say when I have sensitive data that needs to be shared via email I use PGP but other than that I have enough trust in Microsoft and Google... Misplaced? Maybe.
My point was less about whether or not they should be trusted, but that unless you're running your own email server, it's a third party, by definition :)
