It sounds to me like you have a rain water leak someplace, if you have the car outside and rain water may have come in.
It is odd though that it would start to leak out only because you started the engine, and did not move the car or do something else, although just sitting in it may have shifted the weight enough for the water to start flowing out.
Either way I would get to the bottom of this, since standing water is a recipe for disaster (rust).