If it is under the water pump, then it is likely that the water pump is leaking itself. When the car is running, there is a little less pressure being exerted on the any one spot in the cooling system than there is when you turn it off after it's warmed up. It is a basic rule of physics and fluid dynamics. So it leaks more while sitting and not running. Also, while you are driving, wind/air is being blown at it and dispersing any fluid that does come out, so it does not appear to leak. It probably is leaking while running though.
Option A: My suggestion to you is to borrow a pressure tester for the cooling system from the local car parts store (AutoZone, Advance Auto, Oreillys, etc.). With the vehicle off and cold, take the radiator cap off, hook up the pressure tester, and pump it up to 16 PSI. Do not put more than 16 PSI on it, as that could damage things. Then follow the hissing sound to your leak.
Option B: is to fill the cooling system as much as possible. With the cap still off of it, start it with the heat on full blast and let it idle a few minutes while you keep pouring water in it. When it appears to not be taking any more water, turn the engine off, put the radiator cap back on, and then start it again. Watch closely for the leak. Follow it with a flashlight if possible. But be careful because the electric cooling fan can and will come on without warning.
Option A, is the more professional way to do it. Option B, is the more backyard way. Go with Option A if at all possible.
Here is a more in-depth guide: https://www.2carpros.com/articles/radiator-pressure-test
Let us know and we'll go from there.
Monday, July 3rd, 2017 AT 7:35 PM