Learn English – What {do / does} this and that have in common?” — which

As snailplane and Man_From_India tell you, your question does not involve adjectives, which never have a distinct plural form, but attributive nouns.

The singular form is certainly the ‘default’ for attributive constructions, but plural attributives are not uncommon. Some of these (and probably most of the older ones) come about because the singular and plural forms have different meanings. For instance

• We take an ‘arts degree’ or a ‘humanities degree’ because the singular would imply something different: an ‘art degree’ is a degree in painting or sculpture or something of the sort rather than a degree from the School or Faculty of Arts, and a ‘humanity degree’, would be a degree from the Department of Humanity, if such a thing existed.
• We speak of the ‘civil rights movement’ because ‘civil rights’, plural, is a term with a specific meaning, the entire body of rights supposed to be held by all members of a civil society rather than a specific civil right or a general desire for ‘rightness’ in civil society.

In other cases, the plural seems to have come about as a sort of compromise where the underlying sense might represent a simple plural or a singular possessive or a plural possessive—see, for instance, this blogpost on the correct spelling of Veterans Day. (But this is by no means a rule; see this Google Ngram on various names for the laws under which US workers are compensated for on-the-job injuries.)

In yet other cases, grammatical concord seems to be in play. We speak of Virginia Woolf as a woman writer, but we are more likely to call Woolf and Eudora Welty and Flannery O’Connor women writers than woman writers. Geoffrey Leech, in Change in Contemporary English, Cambridge, 2009 (220) suggests that this is more likely when the plural is irregular, without an -s.

But none of this explains why plural attributives became much more common in the second half of the 20th century, particularly in British English. Leech suggests that:

The increase of plural attributive nouns may well have been brought on by the general increase in noun sequences, bearing in mind that the implicit link between two adjacent nouns has to be inferred, putting a burden on the cognitive processing abilities of the reader. In such circumstances an -s at the end of a word may be a handy clue to interpretation. Thus the generic meaning of the plural in weapons purchases or fisheries protection is helpful in eliminating the possibility of a singular interpretation (that one weapon is to be purchased, or that one fishery is to be protected) and in capturing the intended generic meaning of the whole (220-221).

Leech says that Stig Johannson, Plural attributive nouns in present day English, Univ. Lund, 1980, identifies more factors, but this does not appear to be available free online.

What do her parents do?

Here the subject of the sentence is her parents. Because her parents is plural the auxiliary verb DO must agree with the plural noun phrase, so we need do and not does. The auxiliary verb DO is the first verb in the sentence. This is the verb that moves in front of the subject. It has no meaning, it just helps to make the sentence a question.

The verb after auxiliary Do is ALWAYS an infinitive. It can never be "Xing", "Xs", "Xed" or "to X".:

• Does he eats?
• Did they went?
• We don't liking...
• They didn't to come.

It should be:

• Does he eat?
• Did they go?
• We don't like ...
• They didn't come.

We can only have one auxiliary verb DO in a sentence.

However, the second DO in the Original Poster's example is the main verb. It's the lexical verb DO. It isn't an auxiliary. Because it comes after the auxiliary DO, it must be in the infinitive. The verb after auxiliary DO is ALWAYS an infinitive. Therefore the sentence must be like this:

• What do her parents do?

Hope this is helpful!