What is the شرعي ruling regarding the wife's place of residence, given that the husband is traveling for work outside Saudi Arabia, and there is only the marital home, and the wife's father is deceased, and her mother is married to a sick man?

A Muslim is required to work and earn a lawful living, as there is an encouragement for work and lawful earning. If you do not find work in your country but find it in Saudi Arabia or other Muslim lands, there is no religious impediment to going there to work, either accompanied by your wife or leaving her with her family. Seeking work is indeed your right, to help your parents, your siblings, and your wife's family, for the upper hand is better than the lower hand.