Is the father obligated to purchase the 'aqiqah, or is it sufficient for the mother of the wife to buy it for the firstborn?

The 'Aqiqah is not obligatory, but rather a confirmed Sunnah. It is required from the one responsible for the child's expenses. If the wife's mother purchases the 'Aqiqah with the father's permission, it is valid. Even if it is without his permission, it is still valid according to some scholars.