What is the ruling on remaining in a non-Islamic country, given the availability of schools and mosques, when the wife desires to live in an Arab country for the sake of raising the children, and the husband wishes to settle in the non-Islamic country to serve his parents?

Residing in disbelieving countries without a valid necessity is not permissible. Therefore, if your marriage to this person necessitates your residing in that country, then it is not permissible for you to accept, because that would entail falling into disobedience to Allah Almighty.