Is the husband obligated to let his wife work outside the home, in compliance with his parents' request, or should he adhere to the agreement made between him and his wife that she dedicate herself to the home and raising the children?

You are not obligated to obey your parents in permitting your wife to work outside the home, because obedience to them is in matters from which they benefit and which do not harm the child. Nevertheless, you must honor them and treat them with kindness. If your wife's going out to work does not entail a Sharia violation, then there is no harm for you in allowing her to do so if she wishes to work, and this is considered an act of righteousness towards your parents.