Solution 1:

Word collocation. Words get paired and their order becomes fixed. Also known as irreversible binomials, binomials and Siamese twins.

It's like bread and butter. You never say those words the other way round, do you?

Likewise:

  • Ladies and Gentlemen

    Mum and Dad

    cats and dogs

    black and white

    backwards and forwards

    thunder and lightning

    peace and quiet

Some pair words can be joined with "or" for instance

  • rain or shine

    now or never

    sooner or later

    right or wrong

    more or less

    all or nothing

Interestingly, in Italian the word order is sometimes the reverse:

  • bianco e nero (white and black)

    cani e gatti (dogs and cats)

    avanti e dietro (forwards and backwards)

    vivo o morte (alive or dead, which if you think about it, makes more sense.)

but always

  • mamma e papà (mum and dad)

Solution 2:

Actually, @JohnLawler, it's not that complicated. According to footnote 4 in that article you linked to:

This freeze [mother and father] points up the place 1 position of mother, found also in such freezes as ma and pa. We believe that mothers are special.

That is, the English language has frozen the relation "mother" in the first position of any idiomatic phrase, and the "natural ordering" of such relations is fixed. Mother always comes before other family members in a pairing because, well, English idiomatic rules think Mom is great.

Solution 3:

OP states:

I'm pretty sure "Mom and Dad" is standard in English.

Without commenting on 'who comes first', I'd like to point out that the term Mom is not common in British English, where the usual term is Mum, as in Mum & Dad or Mummy & Daddy.

I don't know what terms are more common in other parts of the English-speaking world that have had more of a British than American influence (such as the Antipodes).

Solution 4:

Just to try to add something to the conversation, I do believe the common word order in English when using a compound subject when the subject is yourself and someone else is to give the other person the courtesy of coming first, and so far as I know it is quite incorrect to do it the other way around.

E.g. "Peter and I are coming for a visit."

In Latin, apparently a language where false modesty is frowned upon, it is the other way around. During the reign of Henry VIII of England, Cardinal Wolsey wrote a document in which he used the phrase "Ego et rex meus" -- which is literally "I and my king". While some have complained that Wolsey put himself before the King because of his boundless ego (and he was truly an arrogant and egotistical man), it is actually the correct way to write it in Latin! If he had written in English he would have had to write "My king and I".