Step 1: Use the simple pairing rule for pharyngeal pouch derivatives. The 3rd pouch produces the thymus plus the inferior parathyroid, while the 4th pouch produces the superior parathyroid (and the ultimobranchial body).
Step 2: A common point of confusion is that the inferior gland comes from a lower-numbered pouch than the superior gland. This crossover happens because the thymus pulls the inferior parathyroid far caudally during descent, so although it starts from the 3rd pouch it finishes in a more inferior position.
Step 3: Matching to the options, the inferior parathyroid is tied to the 3rd pharyngeal arch and pouch, so the first, second and fourth arches are excluded.
Step 4: Hence the inferior parathyroid develops from the 3rd arch.
\[\boxed{\text{3rd pharyngeal arch / pouch}}\]