This may be a better question for r/Etymology or r/AskAnthropology depending upon which piece of the question you are truly concerned with. For the origins of words and the social dynamics of families, respectively.

But, as you asked: As to the origins of "Mother" and "Father", specifically, in English they come from Proto-Indo-European (PIE), the oldest ancestral language we can reconstruct from most european and some non-european languages. Mother comes from PIE *Meħter^[1] , producing things such as Latin Māter, English Mother, and Sanskrit Mātr amongst other things. Father comes from PIE *Pħter^[1] , producing things such as Latin Pater, English Father, and Sanskrit Pitr amongst other things.

To my knowledge, there's not any deeper etymology to these words, even in Proto-Indo European, besides "Mother" and "Father". These are basic relationships of human beings and are necessary primitive vocabulary items.

The PIE family is big, but even non-PIE languages usually have words for mother that sound like 'mama', at least in baby-talk. (But not always! Consider Japanese 'Haha', from Old Japanese 'Papa' for Mother). Such words arise because they're easy for children to make, and the first noises children make are often seen as terms for relatives or attempts to get the attention of relatives. After all, it would be far less likely for the first noise a child makes to be taken as an attempt to indicate "House" or "Astrology", would it not?

Whether one never calls one's parents by name is entirely culturally dependant. A question on AskAnthropology about whether there are any exceptions to that rule and why the rule is so common might get you a better answer. At least in the culture I'm from, it's a sign of respect, in the same way that you call your Aunt "Aunt" or your Older Brother "Older Brother" rather than a mere name. Or, for that matter, why a Doctor is "Doctor" rather than his own name.

[1] We don't have any direct evidence of PIE as it was spoken well before writing and is reconstructed via regular correspondences in related languages' words. As its pronunciation is hypothetical, which we indicate its uncertainty via a prefixed asterisk in Linguistics. Here I have also given the symbol "H₂" used for an unknown consonant as ħ (one of its potential pronunciations) and removed some vowel diacritics, to make the words look less alien.