**Core Concept**
The parotid gland receives its nerve supply from various sources, including branches of the sympathetic and parasympathetic nervous systems. The sympathetic supply is primarily derived from the sympathetic plexus surrounding the external carotid artery, while the parasympathetic supply originates from the auriculotemporal nerve, which carries fibers from the otic ganglion. The greater auricular nerve also provides sensory innervation to the gland.

**Why the Correct Answer is Right**
The facial nerve (cranial nerve VII) primarily supplies the muscles of facial expression and does not directly innervate the parotid gland. Its role is mainly motor in nature, whereas the parotid gland's nerve supply is primarily sensory and autonomic. The auriculotemporal nerve, a branch of the mandibular division of the trigeminal nerve (cranial nerve V), carries parasympathetic fibers from the otic ganglion that stimulate salivation. The sympathetic plexus around the external carotid artery provides sympathetic innervation to the gland, promoting vasoconstriction and reducing glandular secretion. The greater auricular nerve, a branch of the cervical plexus, provides sensory innervation to the gland and surrounding skin.

**Why Each Wrong Option is Incorrect**
**Option A:** The facial nerve primarily supplies the muscles of facial expression and is not directly involved in the innervation of the parotid gland.
**Option B:** The auriculotemporal nerve is indeed involved in the parasympathetic innervation of the parotid gland, making this option incorrect.
**Option C:** The sympathetic plexus around the external carotid artery does provide sympathetic innervation to the parotid gland, making this option incorrect.
**Option D:** The greater auricular nerve does provide sensory innervation to the parotid gland, making this option incorrect.

**Clinical Pearl / High-Yield Fact**
The parotid gland's nerve supply is primarily derived from the auriculotemporal nerve, which carries parasympathetic fibers from the otic ganglion, and the sympathetic plexus around the external carotid artery. The greater auricular nerve provides sensory innervation to the gland and surrounding skin.

**✓ Correct Answer: A. Facial nerve**