Does the parasympathetic system slow the digestive system?
No, the parasympathetic nervous system will speed up digestive tract.
The parasympathetic nervous system activates your body's "rest and digest" pathways - which is associated with increased motility in your digestive tract.
The sympathetic nervous system would slow the digestive tract. Sympathetic responses are associated with the "fight or flight" response - for example, being chased by a Grizzly bear.
During a situation like this, you would want the digestive system to be inhibited so that you could escape!