**Core Concept**
Vocal cord palsy in thyroid surgery is a known complication that can occur due to injury to the recurrent laryngeal nerve, which is responsible for innervating the posterior cricoarytenoid muscle, the only muscle that abducts the vocal cords. This nerve is closely associated with the thyroid gland and can be at risk during thyroid surgery.

**Why the Correct Answer is Right**
The recurrent laryngeal nerve originates from the vagus nerve and has a complex course that passes close to the thyroid gland. During thyroid surgery, manipulation or transection of this nerve can lead to vocal cord palsy, resulting in hoarseness of voice, breathlessness, or difficulty swallowing. The nerve is particularly vulnerable due to its proximity to the thyroid gland and the limited space available for surgical dissection.

**Why Each Wrong Option is Incorrect**
**Option A:** The **external laryngeal nerve** is a branch of the superior laryngeal nerve and primarily supplies the cricothyroid muscle, which is involved in adjusting the pitch of the voice. Injury to this nerve would not typically result in vocal cord palsy.

**Option B:** The **inferior thyroid artery** is a branch of the thyrocervical trunk and supplies the thyroid gland. While injury to this artery can lead to significant bleeding, it is not directly related to vocal cord palsy.

**Option C:** The **parathyroid glands** are located in close proximity to the thyroid gland and can be at risk during thyroid surgery. However, injury to the parathyroid glands would typically result in hypocalcemia, not vocal cord palsy.

**Clinical Pearl / High-Yield Fact**
To minimize the risk of vocal cord palsy during thyroid surgery, surgeons should take great care to identify and preserve the recurrent laryngeal nerve, which can be visualized by gently retracting the thyroid gland and using a nerve stimulator to confirm the presence of the nerve.

**Correct Answer: C. Parathyroid glands.**