Whom should child love, respect, and care for more and why? Or, should he equally love both parents?
closed as primarily opinion-based by sabbahillel, mbloch, kouty, Daniel, rosends Jun 15 '16 at 10:43
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In the section of the 10 commandments, it mentions the father first "honour your father and your mother" with the verb of Kavod - honour.
However in parshat Kedoshim (Leviticus, 19:3), it mentions the mother first where the verb is Tir'u from the root Yirah - fear.
On the second of these, Rashi states:
[Every man] shall fear his mother and his father: Here, Scripture mentions the mother before the father, because He is privy to the fact that a child fears his father more than his mother [and therefore, by mentioning the mother first, Scripture emphasizes the duty of fearing her also. However,] in the case of honoring [one’s parents], Scripture mentions the father before the mother, because He is privy to the fact that a child honors his mother more than his father, since she wins his favor by [speaking kind and loving] words. [Therefore, by mentioning the father first in the context of honor, Scripture emphasizes the duty of honoring him also]. — [Kid. 31a]
Other laws, e.g. death for cursing or striking a parent, applies equally to the father and the monther. Similarly the laws of mourning apply equally for a mother as to a father.
Therefore they are treated equally in halacha.
There are some roles however that are specific to the father. If the child is a boy, it is the responsibility of the father, not the mother, to arrange Bris Milah, and if he's a first born then Pidyon Haben, and also the responsibility of the father to arrange the son's education, both Torah and Parnasa, and to find him a bride. (Gemara: Kiddushin, around Amud 29)
That doesn't make him more "important". They are just his roles.