My parrot talks gruffly. Will he be able to learn new words from my daughter?

This should not be a problem, although you and she may have to be patient, because parrots, rather like people, find it easier to learn speech at a relatively young age.
Speaking clearly and slowly should encourage the bird to pick up new words and phrases. The timespan required varies - depending both on the individual parrot, and the length of time which your daughter spends teaching her pet.

Since parrots mimic sounds, so they can acquire a vocabulary based on the speech of a number of people. In fact, some of these birds may even speak in several languages.
You might find that your parrot adds your daughter's words on to the end of phrases which it has already learnt at first, making it sound as if two different people are speaking together. With further encouragement, it should then start to repeat these words on their own with little prompting.

Click for here for more information about talking birds.