Why does my 94 Honda Accord cooling fan only operate when the car is turned off?

The temperature gauge is fluctuating into the high temp area when the car is idling or in stop/go traffic. When I drive, the temp cools down. I've installed a new thermostat and performed a radiator flush and refill. Looking under the hood at idle, the radiator cooling fan does NOT come on as the engine temp increases, but DOES come on after I turn the ignition off. Are we talking a faulty temperature coolant sensor here? Why does the fan come on only after the engine is off?

IMPORTANT: The fan DOES operate (along with the compressor fan) when the A/C is turned on, and that seem to cool the engine down.

The new thermostat may be defective. In the past I've already had times where I got 3 thermostats before i got a good one. STANT thermostats are the only ones I buy now.
Did you get all of the air out of the cooling system?
If there's an air pocket it can do exactly what you describe is happening.
Take the radiator cap off and run the motor for 15 minutes with the climate control set to the hottest setting. The insures that you get the air out of the heater core too.
You may want to go to a mechanic and have a 'block test" done if there's any air in the cooling system. You nay have a head gasket starting to leak.

No idea why it comes on after cutting the engine off, but you have two fans. One is for cooling, one is for A/C. Your A/C condenser needs air flowing to work properly, therefore turning A/C on helps to cool the engine when the radiator fan is not working.
You should find out which fan is actually coming on after turning the ignition off. That would help you track it down easier. If you wanted to try replacing a cheap switch, it should be labeled "fan switch" if you go to a parts store. The problem is once you take the old one out, you'll leak some coolant, so you'll need to bleed your cooling system afterward. If you're not familiar with that procedure you'll save a lot of headaches just taking it to a shop for diagnosis and repair.

Did you put the correct thermostat for your car because diffrent thermostat open at a diffrent temperature also your cooling fan comes on when the car off because the engine is off and their no water going thru it or oil splashing around to decipate the heat so your engine gets about 5 to 10 degrees hoter with engine off and your cooling fan is suppose to come on with engine off