The hormone that primarily increases blood calcium levels to maintain calcium homeostasis is the Parathyroid hormone (PTH).

Parathyroid hormone is secreted by the parathyroid glands in response to low blood calcium levels.
Its main function is to raise and maintain blood calcium concentrations within a narrow range. It achieves this by:
- Stimulating osteoclast activity, which breaks down bone matrix and releases calcium into the bloodstream.
- Increasing calcium reabsorption in the renal tubules, reducing calcium excretion in urine.
- Enhancing activation of vitamin D (calcitriol) in the kidneys, which increases intestinal absorption of calcium.

In contrast:
- Calcitonin lowers blood calcium levels by inhibiting osteoclast activity.
- Calcitriol (active vitamin D) primarily increases calcium absorption from the intestine but works synergistically with PTH rather than independently regulating blood calcium.
- Thyroxine (T4) is involved in metabolic regulation and does not directly influence calcium homeostasis.

Thus, the primary regulator that increases blood calcium levels is Parathyroid hormone (PTH).

Reference: Guyton and Hall Textbook of Medical Physiology, 14th Edition, Chapter 64: Parathyroid Hormone, Calcitonin, and Vitamin D