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: 
To answer your specific question, *Can you have **does** and **do** in the same sentence like the first one?*, Yes, you can. You can have *do* twice, but not *does* twice. This is not a very helpful rule, however. The role that each word plays is very different in each case.

The first use of *do*/*does* is just a way to construct a question in English. It has nothing to do with the second use, which is what the question is really about. Try with a different verb in the question:

*What **do**/**does** her parents **cook**?*

Clearly, the word *cook* is irrelevant to whether we should use *do* or *does*. In reply to your hypothetical question, you could preserve the *do* to give a more assertive tone. 

*Her parents **do** cook.*

*Her parents **do** do.*

Or in singular form:

*Her father **does** cook.*

*Her father **does** do.*

So there's nothing wrong with a sentence with multiple uses of *do*.
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.