Should a young man travel to seek a livelihood and build his future despite his fear that his parents, who are attached to him, may be afflicted with harm?

There is no harm in traveling if the mother is under the care of her children, and you are in need of travel, and the travel does not prevent you from honoring your mother. For honoring one's mother has multiple doors and means. However, it is advisable not to travel and to remain with the mother and family if there is no strong need for travel. For the son's presence with his mother and his closeness to her, along with his honoring of her, is among the greatest reasons for Allah's pleasure. Indeed, Uwais Al-Qarni missed the companionship of the Prophet, peace be upon him, due due to his honoring of his mother.