Would I be sinful for traveling for work while my father is ill, given that there are my siblings to care for him?

It is permissible to travel for work even if the parents do not give permission. If the father gives permission, then it is even more permissible. There is no blame upon you as long as your brothers are taking sufficient care of your father. It would be good if you could afford to pay for a servant for them, to assist them in serving him.