Insights:

The parathyroid glands, thymus and C cells of the thyroid all arise from pharyngeal pouches.

The inferior parathyroids and thymus have their location because of the inferior migration of the 3rd pharyngeal pouch. The superior parathyroids come from 4th pharyngeal pouch. The C cells of thyroid come from the ultimobranchial body of 4th pouch.

Parathyroids and C cells work together in maintaining calcium homeostasis.

Thymus is the site of central tolerance of developing T lymphocytes.

It is clear how abnormalities in pharyngeal pouch development can lead to both endocrine and immune problems.

Reference

Johansson, E. et al. (2015). Revising the embryonic origin of thyroid C cells in mice and humans. Development, 142(20), pp.3519-3528.