A thorough academic accreditation process vets an international school in ways parents cannot, even if they currently have children in the school. That process, along with other processes, forces a school to pull up its socks and closely examine its own standards and plans.
It allows an independent body to look behind the scenes for appropriate governance and clean bookkeeping and encourages transparency– which is particularly important in international schools, when many are privately owned and, according to the laws of the land, it is quite legal for an owner to do whatever they want..
Accreditation also matters because without it, qualifications are not usually accepted by universities and (in the US) college credits cannot be transferred; nor are degrees accepted for graduate schools. There are exceptions to this; occasionally universities will overlook a questionable or non-existent accreditation problem if the student is enormously exceptional), but mere mortals do need to come from a school with known academic qualifications such as Cambridge International or International Baccalaureate.
The real question for a school without any legitimate academic accredition is, ‘Why not?’ A good international school will offer an internationally recognized academic programme.