Start by translating the lab values. A calcium of 5 mg/dl is clearly low and accounts for the neonatal seizures, the phosphate of 9 mg/dl is high, and the PTH of 30 pg/ml sits in the middle of the stated normal range. The diagnosis must explain all three findings together.
Use the calcium and phosphate together first. Low calcium with high phosphate is characteristic of a parathyroid hormone effect that is missing at the tissue level, because PTH normally lowers phosphate by promoting its renal excretion. Vitamin D deficiency would instead give a low or low-normal phosphate, so it does not fit.
Now bring in the PTH. If the parathyroid glands were simply failing, the PTH would be low. Instead it is normal, which is actually inappropriate, because such severe hypocalcaemia should be driving PTH sky-high. A normal PTH that cannot correct the calcium indicates resistance of the bones and kidneys to PTH, which is the definition of pseudohypoparathyroidism. Hyperparathyroidism is excluded by the low calcium, and hypoxic ischaemic encephalopathy does not create this biochemical triad.
\[\boxed{\text{Pseudohypoparathyroidism}}\]