- Vacuum leaks: Check for leaks in the vacuum system, particularly around the intake manifold and hoses. A vacuum leak can cause the engine to run lean and cause the engine to shake while in park.
- Ignition issues: Faulty spark plugs, wires, or ignition coils can cause the engine to run rough and cause the engine to shake while in park.
- Fuel injector problems: A clogged or faulty fuel injector can cause the engine to run rough and shake while in park.
- Engine mounts: Worn or broken engine mounts can allow the engine to move excessively, causing it to shake while in park.
Since you have recently had work done on the cooling system, it is also worth checking the following:
- Coolant level: Ensure that the coolant level is correct. If the coolant level is low, the engine may overheat and start shaking.
- Thermostat: A faulty thermostat may cause the engine to run at an incorrect temperature, leading to shaking.
- Radiator cap: Ensure that the radiator cap is properly tightened. If the radiator cap is not closed properly, it can lead to overheating and shaking.
Check these items and let me know if you need further assistance.