**Core Concept**
The sensory supply of the larynx below the vocal cord is provided by the recurrent laryngeal nerve, which is a branch of the vagus nerve (cranial nerve X). The recurrent laryngeal nerve is responsible for transmitting sensory information from the laryngeal mucosa, including pain, temperature, and touch.

**Why the Correct Answer is Right**
The recurrent laryngeal nerve arises from the vagus nerve, which is located in the neck, and then descends through the thoracic cavity before ascending back up to the larynx. As it returns to the larynx, it provides sensory innervation to the mucous membrane below the vocal cords, including the trachea and the esophagus. This innervation is essential for protecting the airway and preventing damage to the surrounding structures.

**Why Each Wrong Option is Incorrect**
**Option A:** This option is incorrect because the internal laryngeal nerve, a branch of the superior laryngeal nerve, provides sensory innervation to the laryngeal mucosa above the vocal cords, not below.

**Option B:** This option is incorrect because the external branch of the superior laryngeal nerve is responsible for innervating the cricothyroid muscle, which is involved in adjusting the pitch of the voice, not providing sensory innervation.

**Option C:** This option is incorrect because the phrenic nerve provides motor innervation to the diaphragm, which is involved in respiratory function, not providing sensory innervation to the larynx.

**Option D:** This option is incorrect because the glossopharyngeal nerve (cranial nerve IX) provides sensory innervation to the oropharynx and the tongue, not the larynx below the vocal cords.

**Clinical Pearl / High-Yield Fact**
The recurrent laryngeal nerve is a key structure that can be damaged during thyroid or parathyroid surgery, leading to vocal cord paralysis.

**Correct Answer:** C.