The hormone that regulates blood calcium and phosphate levels in the human body is option 3: Parathyroid hormone (PTH).

Parathyroid hormone is produced and released by the parathyroid glands, which are four small glands located behind the thyroid gland in the neck. PTH plays a crucial role in maintaining the balance of calcium and phosphate in the blood. Here`s a detailed reasoning for why parathyroid hormone is the correct answer:

1. Function: Parathyroid hormone acts on various organs and tissues to regulate calcium and phosphate levels in the blood. It has several effects, including:

- Increasing calcium reabsorption: PTH enhances the reabsorption of calcium in the kidneys, reducing its excretion in urine.

- Stimulating calcium release: PTH promotes the release of calcium from bones into the bloodstream.

- Increasing phosphate excretion: PTH inhibits the reabsorption of phosphate in the kidneys, leading to increased phosphate excretion in urine.

- Activating vitamin D: PTH indirectly stimulates the activation of vitamin D, which is essential for calcium absorption in the intestines.

2. Counteracting low blood calcium: When blood calcium levels drop below the normal range (hypocalcemia), parathyroid glands release more PTH into the bloodstream. PTH acts to increase calcium levels by stimulating the release of calcium from bones and increasing its absorption from the intestines.

3. Interaction with vitamin D: PTH and vitamin D work together to maintain calcium homeostasis. PTH promotes the production of active vitamin D (calcitriol) in the kidneys, which enhances calcium absorption in the intestines. Vitamin D, in turn, suppresses PTH synthesis and release when blood calcium levels are sufficient.

4. Feedback mechanism: The release of PTH is regulated by a negative feedback mechanism. When blood calcium levels rise, it inhibits the secretion of PTH, preventing excessive release of calcium from bones.

In contrast, the other options do not directly regulate blood calcium and phosphate levels:

- Glucagon (option 1) is a hormone released by the pancreas and primarily involved in raising blood sugar levels. It stimulates the liver to release stored glucose into the bloodstream but does not play a significant role in calcium and phosphate regulation.

- Growth hormone (option 2), also known as somatotropin, is produced by the pituitary gland and promotes overall growth and development. It does not have a direct effect on blood calcium and phosphate levels.

- Thyroxine (option 4) is a hormone produced by the thyroid gland and is involved in regulating metabolism. While thyroid hormones have indirect effects on calcium and phosphate metabolism, they do not primarily regulate their levels in the blood.

Therefore, the hormone that specifically regulates blood calcium and phosphate levels is parathyroid hormone (PTH).