Step 1: Recall the hypothalamo-hypophyseal axis.
Oxytocin and vasopressin (ADH) are actually synthesized by neurosecretory cells in the hypothalamus, but they are only stored and released into the blood from a different gland further down the axis.
Step 2: Separate synthesis from release.
The anterior pituitary makes its own set of hormones (growth hormone, TSH, ACTH, etc.) using its own glandular cells, so it can be ruled out here. The adrenal gland is unrelated to this axis entirely.
Step 3: Identify the release site.
The hypothalamic neurons send their axons down the pituitary stalk and release oxytocin and vasopressin directly into the blood from the posterior pituitary (neurohypophysis), which acts purely as a storage and release site.
\[ \boxed{\text{Posterior pituitary}} \]