David moved his parents to Moab in order to protect them from Saul. Being well advanced in years, they would not have been able to endure the life of a fugitive.1 If they were outside of Saul’s territory, then Saul would not be able to harm them directly. With Ruth being David’s great-grandmother, there would have been some natural family connections with the people of Moab. Quite possibly David made use of these connections in his request to the king of Moab,2 and the king granted his request because he was an enemy of Saul and supportive of a split that would weaken Saul.3 David would later fight against Moab and defeat them in battle (2 Samuel 8:2).
3 And David went from there to Mizpeh of Moab. And he said to the king of Moab, “Please let my father and my mother stay with you, till I know what God will do for me.”