Is it permissible to leave a sick father with a brother and return to one's wife and children, knowing that the father was living with me but could not endure the cold weather? And will returning home after leaving the father make me undutiful to him? And will staying with the father make me negligent in my duties toward my wife and children?

Righteousness towards parents is obligatory and is not limited to constant companionship, especially if they do not need it or if they have someone to serve them. Their righteousness includes bringing good to them and warding off evil from them, according to one's ability. If you live with your family and regularly check on your father, visit him, and provide him with financial support, then you are dutiful towards him and not disobedient. However, if your continued stay with your father would lead to neglecting the rights of your wife and children, then you must combine righteousness towards him with fulfilling their rights. This means living with them and checking on your father and visiting him as much as you are able.