If your vehicle is showing engine overheats only when the a/c is running, here are the most likely causes and the parts you'll need to fix each one.
Most modern vehicles use electric cooling fans that are programmed to turn on automatically when the A/C is running to pull air through the hot condenser and radiator. If a fan motor, relay, or control module fails, the engine cannot shed the extra heat, leading to overheating at idle or low speeds.
Dirt, bugs, leaves, or road debris can pack between the A/C condenser and the radiator. The A/C condenser gets very hot when running; if air cannot flow through it to reach the radiator, the engine will overheat under the added thermal load.
On rear-wheel-drive vehicles and trucks with mechanical fans, a thermal fan clutch pulls air through the radiator. If the fluid inside the clutch leaks out or degrades, it won't engage fully to handle the extra heat load of the A/C system at low speeds.
The cooling system may already be operating at its absolute limit due to a partially stuck thermostat, weak radiator cap, or slightly low coolant. Turning on the A/C adds just enough extra heat to push the temperature gauge into the red.
If the A/C compressor is failing internally, it can create massive mechanical resistance. This forces the engine to work much harder to turn the serpentine belt, generating excess heat that overwhelms the cooling system.
If the cooling fans were recently replaced or rewired, they might be pushing air out the front of the grille instead of pulling it into the engine bay. This fights natural airflow while driving, causing overheating specifically when the fans are forced on by the A/C.