How can a Muslim achieve "Purify your earnings and your supplication will be answered" if his parents favor him over his brother in financial giving or food, especially if the brother is negligent in honoring his parents?

You are not required to investigate whether your brother received the same amount as your father or mother gave you. It is permissible for you to accept your parents' preference for you, as it is not considered contrary to justice among children. It is permissible for a parent to withhold a gift from a child due to their wickedness (fisq), and disobedience (uquq) is one of the greatest forms of wickedness. There is no disfavor in a parent preferring their dutiful child over a disobedient one.