Question: > 
> * What does her parents do?
> * What do her parents do?
> 
> 
> 

Which one is correct? Can you have *does* and *do* in the same sentence like the first one? Would it be incorrect because parents is plural so *do* must be used throughout?

Answer: 
> 
> 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!
The correct question is, "What do her parents do?"
Since the word 'parents' is plural you would use 'do.' 
Of course if you had the word 'parent' instead of 'parents'you would use does.
"What do her parents do?"
"What does her parent do?"