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.
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?"