The explanation seems to be that these names are true genitives, and that John Parsons was John the Parson's man, while John Monks was employed by the monastery.
"The Romance of Names"
Ernest Weekley
They predicate something of the son or of the father; and if we distinguished between the sons of the father, and the sons of the mother, the genitives would mark the class or genus to which the sons respectively belonged.
"Lectures on The Science of Language"
Max Müller
This is the true explanation of all the "double genitives" which our grammarians discover; for when the first noun is partitive, it naturally suggests more or other things of the same kind, belonging to this possessor; and when such is not the meaning, this construction is improper.
"The Grammar of English Grammars"
Goold Brown