Answer
Calcitonin is secreted by the parafollicular cells of the thyroid gland. Parathyroid hormone (parathormone) is secreted (normally) by the chief (principal) cells of the parathyroids.
Work Step by Step
Both the follicular cells and the parafollicular cells of the thyroid gland secrete important hormones. The thyroid hormones (T3 and T4) are produced by the follicular cells; the other hormone , calcitonin is secreted by the parafollicular cells which are located between the follicles.
In the case of the parathyroid glands, there are also two types of parenchymal cells. The chief or principal cells, and th oxyphil cells. Parathormone is normally produced by the chief cells only. However, it is known that in cases of parathyroid gland cancer, the oxyphil cells also secrete parathormone. The metabolic purpose for this is unclear.