**Core Concept**
The autonomic nervous system (ANS) is a branch of the nervous system that controls involuntary functions, such as heart rate, digestion, and respiratory rate. The ANS has two main subdivisions: the sympathetic nervous system (SNS) and the parasympathetic nervous system (PNS). The PNS is responsible for promoting rest, digestion, and other 'housekeeping' functions.

**Why the Correct Answer is Right**
The PNS is composed of preganglionic and postganglionic neurons. Preganglionic neurons originate from the central nervous system (CNS) and synapse with postganglionic neurons in peripheral ganglia. The PNS has several cranial nerves and spinal nerves that carry parasympathetic fibres. However, the trigeminal nerve (cranial nerve V) is unique in that it does not have any parasympathetic fibres. This is because the trigeminal nerve primarily carries sensory information from the face and motor information to the muscles of mastication.

**Why Each Wrong Option is Incorrect**
**Option A:** This option is not provided, but in general, the trigeminal nerve's lack of parasympathetic fibres is the correct answer.
**Option B:** The facial nerve (cranial nerve VII) does have parasympathetic fibres that control salivation, tear production, and other functions.
**Option C:** The glossopharyngeal nerve (cranial nerve IX) carries parasympathetic fibres that control salivation and other functions.
**Option D:** The vagus nerve (cranial nerve X) is a major nerve that carries parasympathetic fibres to the thorax and abdomen.

**Clinical Pearl / High-Yield Fact**
The trigeminal nerve's sensory branches are responsible for facial sensation, including pain, temperature, touch, and proprioception. It is also the nerve responsible for corneal sensation and the blink reflex.

**Correct Answer: A. The trigeminal nerve.**