You can partner up by sending an invite to your Facebook or mutual-follow friends, Suggested Partners, or by using the Invite link.

Open the Co-Op Event, then go through the prize and rules pages to the Pick a Partner page, where you can send invites.
If you have an open partner slot, tap the Add a Partner button on the main event page to bring up the same page where you can send invites.

After you send out invites, recipients will receive a notification that they have a pending request to accept. Once an invite has been accepted, the event begins!