Drooling does not occur because more saliva is produced but because, without realizing it, the natural tendency to swallow every now and again (even when not eating) is slowed down. This, combined with stooped, poor posture and an inability to close the lips as normal, affects the control of the saliva in and from the mouth.
As a result saliva tends to accumulate in the mouth, and rather than being swallowed it can overflow. Many people have poor lip closure due to their Parkinson’s and a flow of saliva from the mouth will occur if the lips do not seal tightly. There is also a tendency for the head to be bent forwards so that the chin points downwards in to the chest. This will make the drooling worse as swallowing is impaired by the tucked-in chin. Try for yourself. It is very difficult to swallow if your chin is tucked well in to your upper chest.