Could you give a source for the "well known" effect of salt to cause hyperparathyroidism?
In rats, consuming NaCl raises blood ionized calcium levels, most likely due to a direct physicochemical interaction of sodium with calcium-binding blood proteins. This elevation in blood ionized calcium is coincident with, and probably causes, lower PTH levels. See details in: Tordoff MG. NaCl ingestion ameliorates plasma indexes of calcium deficiency. Am J Physiol Regul Integ Comp Physiol 273: R423-R432, 1997.
High salt intake is associated with higher urinary calcium excretion. The path however from increased calciuria to secondary hyperparathyroidism is quite long.