Is it permissible for a brother to travel to perform Umrah for the first time, while his father is sick and needs his care and to be carried when he falls?

The preponderant opinion is that 'Umrah is obligatory once in a lifetime for those who are able. If the father is in need of his son's constant presence and there is no one else to care for him, then the son is not permitted to travel for 'Umrah, because his absence would cause harm to his father. This is an exceptional case to the general rule that parental permission is not required for individual obligatory duties such as Hajj, 'Umrah, and prayer.