Streaming to twitch involves two parties. First, the service provider twitch, second the user uploading the video. Streaming to a second service introduces a third party.
"Third party" just means anyone who isn't a party to the contract. They're called that because most contracts have two parties (e.g., buyer and seller, landlord and tenant, service provider and customer). Contracts can have three or more parties, but they're not usually numbered. The TV trope of incomprehensible legalese involving "the party of the first part" and "the party of the second part" is a joke that you won't actually find in modern agreements.