Step 1: Read both statements.
Statement I links the parasympathetic nerves to gut movement. Statement II says passing out faeces is involuntary. We test each one.
Step 2: Recall the autonomic nerves.
The parasympathetic system is the "rest and digest" part of the nervous system. It boosts digestion when the body is calm.
Step 3: Judge statement I.
Parasympathetic stimulation speeds up peristalsis, the wave-like squeezing that pushes food along the gut. So statement I is correct.
Step 4: Understand egestion.
Egestion is the removal of faeces through the anus. The rectum has an internal sphincter that opens by reflex, but there is also an external sphincter.
Step 5: Judge statement II.
The external anal sphincter is under our conscious control. So the final act of passing faeces is voluntary, not involuntary. Statement II is wrong.
Step 6: Pick the option.
Statement I is correct and statement II is incorrect, which is the third option.
\[ \boxed{\text{Statement I is correct, but Statement II is incorrect}} \]