父 「Fù 」means Father. 母 「Mu 」means Mother. As its name suggests,
this palace has everything to do with one's parents. It covers their general
luck, martial bliss, health and longevity and to a certain extent, their character.
The karmic relationship with one's parents is also indicated here. Whether
a lot or little help comes from them (help could be in monetary terms or
advice), or the parents-child relationship is in good or bad shape, all
these fall under karmic relationship.
There is a Chinese saying, 一日为师，终身为父 「Yī Rì Wéi Shī, Zhōng Shēn Wéi Fù 」,
which means, a teacher is respected and treated as you would a father. So, one's
relationship with his teachers and mentors can be indicated in 父母宫 as well.