Does a son's travel without the knowledge of his father, who is ill with Alzheimer's, constitute disobedience, especially if the travel is necessary to cover the father's treatment expenses and living costs, and after performing Istikhara and things were made easy?

It is permissible to travel to seek a livelihood without the parents' permission if the travel is necessary to meet the traveler's or their family's needs, in order to avert harm. However, as for travel for the purpose of accumulation and seeking increase, even if the parents permit it, it is forbidden due to the corruptness of its purpose, as there is no fear of perishing in it. There is no harm in traveling to seek a livelihood and leaving the father with the brother; this is not considered disobedience.