I think that all three are important, although I don't think I ever thought I wanted to be the best but I did always want to play the piano. Your parents are an important part of the process because they should insist you do your practice, after all they are paying for your lessons.
Your piano teacher can inspire you with lots of learning stories because they have been in the exact same situation and pupils will tend to listen to their teacher more than their parents when it comes to learning to play the piano.
The most important part of this equation is you. Whatever you learn in life, you are more likely to succeed if you have the desire to do it. Simply put, if you don't want to learn, the greatest parents and teachers in the world wont do any good because after all the encouragement and tuition, its you that has to do the work.