**Core Concept**
Hypocalcemia is a common complication of thyroid surgery due to accidental damage to or removal of the parathyroid glands. Parathyroid hormone (PTH) plays a crucial role in regulating calcium levels in the blood by stimulating the release of calcium from bones and increasing its absorption from the gut.

**Why the Correct Answer is Right**
In the case of perioral paresthesia and hypocalcemia after thyroid surgery, the best management would be to administer calcium supplements to raise serum calcium levels. This is because the parathyroid glands are responsible for regulating calcium levels in the blood, and damage to these glands during surgery can lead to hypocalcemia. Administering calcium supplements helps to alleviate symptoms of hypocalcemia, such as perioral paresthesia, and prevents more severe complications like tetany.

**Why Each Wrong Option is Incorrect**
**Option A:** Administering vitamin D supplements alone would not be sufficient to manage hypocalcemia, as vitamin D is necessary for calcium absorption but does not directly address the low calcium levels.
**Option C:** Administering calcitonin, a hormone that lowers blood calcium levels, would worsen hypocalcemia and exacerbate symptoms.
**Option D:** There is no evidence to suggest that administering thyroid hormone supplements would help manage hypocalcemia after thyroid surgery.

**Clinical Pearl / High-Yield Fact**
The most common cause of hypocalcemia after thyroid surgery is damage to or removal of the parathyroid glands, which can occur in up to 10% of cases. Early recognition and treatment of hypocalcemia are crucial to prevent more severe complications.

**Correct Answer:** A. Administer calcium supplements.