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: 
It's "do".

Think about the answer to the questions. "Her parents *does*" is incorrect, whereas 

> 
> Her parents do a mean chicken casserole.
> 
> 
> 

Is natural sounding and correct.
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?"