Is there a jurisprudential principle that states: "For the father is obedience, and for the mother is contentment"? And if there isn't, then what is the response to one who says it, and what are the limits of obeying the mother in Islam?

The questioner might be referring to the statement of Imam Malik to someone who asked him about going to his father in Sudan while his mother forbade it. He answered him: "Obey your father and do not disobey your mother." Al-Qarafi interpreted this as prohibiting going out without the mother's permission. There is no jurisprudential rule stating that the father is owed obedience and the mother is owed رضا (contentment/رضا). Filial piety and showing kindness to one's parents are obligatory, and obedience to them is in what is right and pleasing them as much as possible. If it is impossible to combine obedience to both, then the mother's right is more emphasized than the father's, according to the Hadith (definition) of Abu Hurairah: "Your mother, then your mother, then your mother, then your father." Obedience to the mother is in what is right and free from sin, according to the saying of the Prophet, peace be upon him: "There is no obedience in sin; obedience is only in what is right."