What does the Qur’ān and Āhadīth say about the rights/relationship between husband’s parents and wife and in the same way between wife’s parents and husband?
I am afraid there are no categorical injunctions in the Qur’ān that specifically mention the rights of the spouse’s parents. The Book of Allah, however, asserts that you should show kindness to your parents (4:36). What needs to be appreciated is the fact, that marriage is a contractual arrangement that binds two persons into one sacred bond. As an obvious corollary of this agreement, a wife’s relations become her husband’s and a husband’s relations become hers. It is for this reason that I opine that the aforementioned directive of treating your parents well extend to your spouse’s parents as well. You are, indeed, obligated to show kindness to them as you do to your own. What needs to be kept in mind is the fact that you and your spouse are standing in a position of ‘give and take’; the time, care and love that you extend to her family would obviously be returned to yours.