Third party means any person (including companies, partnerships, legal entities, churches, governmental authorities and agencies) who is not a party to the agreement. You could define “Person” in one way or another, as U.S. style contracts sometimes do, but in most (if not all) cases, it looks exaggerated and most likely it does not add anything to the common understanding that if:
a Party shall not engage a third party…
…this means that the Party should do it on its own (and certainly not involve someone else to do it).
Best practice rule. Do not replace third party by “third Person” and do not elaborate on it. If you have a third party in mind, either insert its name in the agreement (and don’t forget to include its affiliates) or mention it for the record in a cover mail with your draft agreement.