Land Rover Repair: 97 Disco SE7 Is Over heating an spilling coolant, waterpump, coolant

Question
I had the car in park and went in the store while my husband sat in car on hot day with AC running for about 20 min. During this time the car started to over heat and the coolant started over flowing out the top. We put more coolant in and when we ran the AC the thermostat read slightly above normal. the next 24 hrs we put two containers of coolant in. There isn't a leak its just over flowing and only when the car is stopped. The thermostat will rise and the shoot back to a little over normal. Could it be the thermostat, or the water pump, or something else?

Answer
This is something to have looked at by a mechanic before it gets worse.  The Rover engine does not tolerate overheating and can be destroyed quickly.  Using coolant is not a good sign.

This could be a bad thermostat, a bad waterpump would likely be constant high temps.