The washer pump for the headlamps is plugged directly into the washer reservoir. The actuation of the headlamp wipers is what triggers the pump to run, and since one of your wipers won't "park", the pump keeps running.

Depending on where the reservoir is, unplugging the pump may or may not be an easy exercise.

You could also try unplugging the wiper motor - if your car is the same as mine, the wiper motor actually plugs into the headlamp housing. That might be an easier solution. Once the motor is disconnected from the electricals in the car, there's no signal to the pump.