Low coolant or water pump, but what car is it? I assume you don't give your daughter your e30.

E39 wiki has plenty of information on the cooling system in the e39

 

If it was driven hard, parked quickly and shut off, heat will build and rise. It did not run long enough to pull the extra heat out of the engine. This would be normal.
If it was driven easy, allowed a little time to idle before this shut off and still had the temp go up you are looking at a problem.
Could be marginal cooling system, low on fluid as mentioned. The heat from driving has to go somewhere and it will always rise to the top.
Find out more before tearing things apart.

 

I agree with Westlotorn. It could be "heat soak"--residual heat in the block, with the coolant not circulating. Check the coolant level to make sure the red indicator pops up. If it's at the proper level, start it up and let it run for a while to see if the temperature is stable. If it's okay at idle, take it out for a short drive and keep an eye on the temperature. It might be fine. You can monitor the temperature on the digital display or using the gauge. If it climbs above 200 degrees F or the needle moves past the 12:00 position, shut it down immediately.

 

Here's an idea - would pressing the "max" button before getting out of the car be an effective way to draw the heat out of the engine and into the car, thereby cooling the engine?


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Is it really overheat?

Do you have an E39? I can't tell ... but if you do ... do you have the high cluster?

See also:
- What to look for when your coolant temperature gauge indicates overheating (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)