Lazarus was a common name in Israel. It means God your help. If Lazarus and the Rich Man is a parable or at least an allegory, then there could be no relationship between Martha and Mary's brother and a fictitious person.
God is your help was a common phrase employed by the Pharisees who claimed there hands were tied so they could not lift a finger to help the poor or disabled without jeoparding their afterlife. Very similar to JW elders who say they must leave unwitnessed alleged child molesters in Jehovah's hands.