Based on the provided error codes (p516 and p235) for a 2002 Ford F-250 with a 7.3L engine, here is a diagnosis of potential parts that need to be replaced, starting with the most likely:
1. **EGR Valve**: The p516 code indicates a problem with the Exhaust Gas Recirculation (EGR) system, which is responsible for reducing NOx emissions. A faulty EGR valve is the most likely culprit.
2. **EGR Cooler**: The p235 code suggests an issue with the EGR cooler, which cools the recirculated exhaust gas before it re-enters the combustion chamber. A malfunctioning EGR cooler can cause the EGR valve to fail.
3. **EGR Temperature Sensor**: This sensor monitors the temperature of the EGR system. A faulty sensor can trigger both p516 and p235 codes.
4. **EGR Vacuum Solenoid**: This solenoid controls the vacuum to the EGR valve. A malfunctioning solenoid can cause the EGR valve to fail, leading to the p516 code.
5. **EGR Position Sensor**: This sensor monitors the position of the EGR valve. A faulty sensor can cause the EGR valve to operate incorrectly, leading to the p516 code.
6. **EGR Tube/Duct**: A clogged or damaged EGR tube can cause issues with the EGR system, leading to both p516 and p235 codes.
7. **EGR Control Module**: The control module manages the EGR system. A faulty module can cause the EGR valve and cooler to malfunction, leading to the p516 and p235 codes.
8. **Intake Manifold Gasket**: A leaking intake manifold gasket can cause vacuum leaks, affecting the EGR system and potentially triggering the p516 code.
9. **Fuel Injectors**: While less likely, faulty fuel injectors can cause misfires and affect the EGR system, potentially leading to the p516 code.
10. **PCV Valve**: The Positive Crankcase Ventilation (PCV) valve can affect the EGR system if it is not functioning correctly, though this is less likely given the specific codes.