Is psuedogout the one associated with CPPD deposits? Maybe those deposits are caused by calcium being sequestered in the deposits, which would account for the hyperparathyroidism. Not sure about hypoPTH or hemochromatosis though.
HypoParathyroidism leads to hyperphosphatemia..this drives calcium in normal tissues and cause damage aka chondrocalcinosis
Normally, parathyroid hormone via proximal tubule Gs (CAMP )cause phosphate excretion.