Final answer:
The parasympathetic nervous system is responsible for stimulating insulin secretion by the pancreas, particularly during rest-and-digest activities.
Step-by-step explanation:
The branch of the autonomic nervous system that stimulates insulin secretion by the pancreas is the parasympathetic nervous system. The parasympathetic nervous system, often referred to as the "rest-and-digest" system, increases gastrointestinal (GI) secretion and motility, which includes the secretion of insulin in response to increased blood glucose levels. This is in contrast to the sympathetic nervous system, which prepares the body for 'fight or flight' situations and generally decreases GI secretion and motility.