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: 
Araucaria answered the part about which is correct. Yes, it's "do", because the subject is "her parents", which is plural, and so requires a plural verb.

As to whether you can use the word "do" twice in a sentence: Sure you can. Maybe you're thinking of the common advice to avoid using the same word twice in a sentence, but this is a matter of style and not an absolute rule. I wouldn't write, "I bought a car from the car salesman at Friendly Car Lot" because the repeated use of the word "car" sounds awkward. But short, common words like "the" and "do" are used repeatedly all the time and no one really notices. 

Note the two "do"s are serving different functions in the sentence. The first is part of the conventional way of phrasing a question about an action: "What do ..." The second is the specific action being asked about. Many other verbs would fit in its place. "What do her parents read?" "What do her parents eat?" etc.
I saw you're struggling between "parent" and parents.

Let's start from your question. First, we need to see the subject here which is "parents", mean both of our mother and father, which means that this is a plural subject. For plural subject, you must use the root word. So,....

What does her parents do? (Wrong)

What do her parents do? (Right)

I saw that you ask Ste about the word "parent". The word "parent" here means one of the parent(father or mother), which is a singular subject. For singular subject, we must add -s, -es, -ies and more. So,.....

What do her parent do? (Wrong)

What does her parent do? (Right)

In conclusion, we must pay attention on the subject to see which is suitable for the sentence.