**Core Concept**
The auriculotemporal nerve is a branch of the mandibular division of the trigeminal nerve (CN V3) that carries parasympathetic secretomotor fibers from the otic ganglion to the parotid gland. These fibers are essential for the gland's secretory function. During parotidectomy, the auriculotemporal nerve is at risk of being injured or reinnervating incorrectly, leading to complications.

**Why the Correct Answer is Right**
In this scenario, the auriculotemporal nerve has reinnervated the branches of the facial nerve (CN VII), which are responsible for the motor innervation of the muscles of facial expression. The reinnervation of the facial nerve by the auriculotemporal nerve leads to aberrant parasympathetic stimulation of the facial muscles, resulting in excessive sweating (hyperhidrosis) on the cheeks during eating. This phenomenon is known as Frey's syndrome or gustatory sweating.

**Why Each Wrong Option is Incorrect**
**Option A:** This option is incorrect because the vagus nerve (CN X) is not directly involved in the innervation of the parotid gland or the facial muscles.

**Option B:** This option is incorrect because the glossopharyngeal nerve (CN IX) is involved in the innervation of the parotid gland, but it is not the nerve that the auriculotemporal nerve would reinnervate in this scenario.

**Option C:** This option is incorrect because the hypoglossal nerve (CN XII) is involved in the innervation of the tongue, not the facial muscles.

**Clinical Pearl / High-Yield Fact**
Frey's syndrome is a classic complication of parotidectomy, and its diagnosis can be confirmed by the presence of gustatory sweating or flushing of the facial muscles during eating. It is essential to recognize this condition to provide appropriate treatment and management.

**Correct Answer: D.** Facial nerve (CN VII)