The Respiratory System is part of what System?

The fact that it is a "system" dictates that it is part of the "respiratory system". But to answer the question I think you are asking I would have to say it works in conjunction with the cardio- pulmonary system in the absorption of oxygen.

Respiration is the act of bringing oxygen into your body and exhaling carbon dioxide and other wastes from your body. This is done via the lungs or pulmonary system. Medicine was first organized in Latin but since this is a language no longer used, we have come up with our own terms as well, which is why we have different names for the same systems (circulatory/cardiac, pulmonary/respiratory, hepatic/liver, renal/kidneys, etc).