Must one obey one's parents and refrain from helping the elderly neighbor, or is there no obedience to a creature in disobedience to the Creator in this case, and what about fulfilling one's promise to the neighbor?

You are not obligated to obey your father in refraining from visiting and caring for your neighbor, unless your father has a valid reason for preventing you, such as fearing temptation or suspicion for you. In that case, you must obey him.