**Core Concept**
Branchial cleft anomalies are congenital malformations that arise from the abnormal development of branchial arches and clefts during embryogenesis. These anomalies can be cystic, fistulous, or sinus-like in nature, and may involve the skin, cartilage, and salivary gland tissues.

**Why the Correct Answer is Right**
The branchial clefts are formed by the invagination of the ectoderm into the branchial arches during embryonic development. Anomalies of the branchial clefts can occur due to disruptions in this process, resulting in cysts, fistulas, or sinuses. The most common branchial cleft anomalies are the second branchial cleft cysts, which typically present as painless neck masses on one side of the neck. These cysts are often located anterior to the upper border of the sternocleidomastoid muscle and can be associated with a sinus tract or fistula.

**Why Each Wrong Option is Incorrect**
* **Option A:** This option is incorrect because branchial cleft anomalies are not typically associated with the parotid gland. While the parotid gland is located in the vicinity of the branchial clefts, it is not directly involved in the development of these anomalies.
* **Option B:** This option is incorrect because branchial cleft anomalies are not typically present at birth or shortly after birth. While some branchial cleft anomalies may present in infancy, many cases are not diagnosed until later in childhood or even adulthood.
* **Option D:** This option is incorrect because branchial cleft anomalies are not typically associated with the thyroid gland. While the thyroid gland is located in the anterior neck, it is not directly involved in the development of branchial cleft anomalies.

**Clinical Pearl / High-Yield Fact**
The most common branchial cleft anomaly is the second branchial cleft cyst, which typically presents as a painless neck mass on one side of the neck. The cyst is often located anterior to the upper border of the sternocleidomastoid muscle and can be associated with a sinus tract or fistula.

**Correct Answer:** C.