First a quick overview of the coolant system on a vehicle - the vehicle's coolant system builds pressure when it is heated (heat causes expansion, which in turns causes pressure). A vehicle's coolant system is made to do this to help keep the coolant from boiling in the engine as the temperatures of an engine do get above the boiling point of water and this pressure in turn raises the boiling point. Once the temperature gets up to operating condition (depending on the thermostat - which is usually around 195 degrees) - the thermostat opens and allows the cooler dexcool/water mixture into the engine and releases the hot mixture out and into the radiator so it can be cooled.
If the pressure gets to high in the coolant system, the radiator cap will allow the pressurized system to vent into the overflow container. (The pressure allowed is determined by the radiator cap - usually around the 13 -15 psi mark). This usually happens if there is some problem in the cooling system - and your vehicle is usually overheating (possibly thermostat sticking or blown head gasket which allows combustion pressures into the coolant system).
Once the vehicle is turned off, the temperature decreases and so does the pressure (do you have a overflow tank and if so, makesure the hose connected isnt clogged, so it can relieve the pressure if needed) - and if any had gone into the overflow tank, it will suck it back in when there is negative pressure on the system through the radiator cap (this is what the little springy thing on the bottom of a radiator cap is for).
If there is no cap directly on the radiator, - then the overflow container is also called an expansion tank and allows the coolant to expand into it. There is a fill line around 1/2 way down that tank so that it will allow for the expansion of the coolant (from the heat).
Usually when the water pump fails - there is a little hole on the bottom which allows coolant out of to signify that the bearing is worn and the seal can no longer seal it - so this determines when the water pump needs replacing.
For your vehicle to leak when the the vehicle after the vehicle is turned off and depressurizing I would think that you will have a small leak that is possibly leaking onto something hot and the coolant is evaporating while running (since the engine would he hot enough). Once the engine is turned off and cools down, the system retains pressure longer than it takes to cool off and at this time it doesn't evaporate and leaks onto the ground.
This would mean that it may be a pin hole in a rad hose above an exhaust manifold, an intake that is seeping slowly onto the engine, possible heater hose, etc.
A person would need more information for a further diagnosis - for example - where is the leak located - front or back of the engine, and if you open the hood after it has leaked - can you see a puddle anywhere above the leak on the engine.