Step 1: Interpret the biochemistry: serum calcium is low at 5 mg/dl (hypocalcaemia, which explains the seizures), serum phosphate is high at 9 mg/dl (hyperphosphataemia), and PTH is within the normal range at 30 pg/ml.
Step 2: The combination of low calcium with high phosphate points to a parathyroid-axis problem rather than vitamin D deficiency, in which phosphate is usually low or low-normal because PTH-driven phosphaturia is intact.
Step 3: In true hypoparathyroidism PTH would be low or inappropriately undetectable. Here PTH is not low; in fact for this degree of hypocalcaemia PTH should be markedly elevated, so a normal value is inappropriately normal, meaning end-organ resistance to PTH.
Step 4: This pattern of hypocalcaemia, hyperphosphataemia and normal or raised PTH defines pseudohypoparathyroidism, where the kidney and bone fail to respond to PTH.
Conclusion: The most probable diagnosis is pseudohypoparathyroidism, so option 1 is correct. Hyperparathyroidism would give high calcium, vitamin D deficiency gives low phosphate, and HIE does not produce this biochemical triad.