Step 1: A rising calcitonin points to medullary thyroid carcinoma, a tumour of the calcitonin-producing C cells, and rising levels after surgery signal relapse or spread.
Step 2: This cancer travels in families as part of MEN 2 syndromes, which bundle MTC with adrenal and parathyroid lesions.
Step 3: Among these, the parathyroid component shows up as hyperparathyroidism in about a fifth to a third of affected patients, so the strongest clinical association of MTC in this list is hyperparathyroidism.
Step 4: Thyroid hormone excess, low parathyroid activity, and cortisol excess do not fit the MEN 2 pattern that drives MTC, eliminating the other three.
\[\boxed{\text{Hyperparathyroidism}}\]