If the parts company were to purchase the aircraft and could show that they owned the plane (IE they were the registered owners in the FAA's mind and for tax purposes, etc), then the parts company could fly their parts around utilizing their own or contract pilots as they saw fit ($ paying their pilots however much the market will bear). This would be a part 91 corporate operation. However they would have to own and insure the plane.
In turn the parts company could lease unused flight time to the flight school in order to better utilize the aircraft and lower the operating costs. During this period, the plane could be insured separately under the flight schools insurance (but only for flight training/rentals).