**Core Concept**
The parotid gland is a major salivary gland located in the preauricular area, and it plays a crucial role in the formation of saliva. The gland is surrounded by several important structures, including nerves, blood vessels, and other glands.

**Why the Correct Answer is Right**
To determine which structure will not run through the parotid gland, we need to consider the anatomy of the area. The facial nerve (cranial nerve VII) runs through the parotid gland, dividing into its terminal branches within the gland. The external carotid artery and its branches, including the maxillary artery and the superficial temporal artery, also pass through the gland. However, the submandibular gland is a separate salivary gland located below the mandible, and it does not run through the parotid gland.

**Why Each Wrong Option is Incorrect**
**Option A:** The facial nerve is a key structure that runs through the parotid gland, so this option is incorrect.
**Option B:** The external carotid artery and its branches pass through the parotid gland, making this option incorrect as well.
**Option D:** Although the submandibular gland is a separate gland, it is not the correct answer because we are looking for a structure that will not run through the parotid gland, and the submandibular gland is indeed related to the parotid gland in terms of salivary gland function. However, the correct answer is not the submandibular gland itself, but rather another structure that is not listed.

**Clinical Pearl / High-Yield Fact**
It's essential to remember that the facial nerve runs through the parotid gland and can be damaged during surgical procedures in this area, leading to facial weakness or paralysis.

**Correct Answer:** D.