The third party must give you at least the airline-issued confirmation number. Until you receive it, your reservation might not be confirmed by the airline, and essentially you might not have a ticket yet. With some third party agents it may take up to 72 hours until the ticket is confirmed by the airline. Call the third party, and ask.
Also the third party usually (but not always) give you the ticket number. If they do not, you can always get it from the airline by telling them the confirmation number.
Please note that you might receive more than one confirmation number for your itinerary if your flights are on different airlines; make sure you use the correct one.
Finally you can call the airline and ask. They are historically hesitant to answer those questions without your confirmation number (they are not sure that you are the person named on the ticket - you might be checking a ticket for someone else), but if they can authenticate you by other means, such as your frequent flier number, they will be able to confirm the issuance of the ticket.