We argue about who should look after our frail elderly parents. How can it be settled fairly?

The duty to care for parents lies with their own children; a spouse helps as a kindness, and the load should be shared fairly among siblings.
Allah commands kindness to parents especially in their old age (al-Isra' 17:23–24). A wife is not legally obliged to care for her in-laws, but her kindness is greatly valued; a husband, for his part, may not put the whole burden of care on his wife so that her rights are neglected (an-Nisa' 4:19). Steps: hold a family meeting with all siblings about rotas, costs and roles; consider a helper, visiting nurses or a day-care centre; and make sure the spouse doing the most caring gets rest. Care for both sides' parents must be weighed fairly. Apply for help from the Social Welfare Department if eligible. See the question on a wife caring for her in-laws.