Is my fast valid if I drank water believing that dawn had broken, then it became clear that it was the Adhan of another mosque before the Adhan of the neighboring mosque which ascertains the correct time? And should I obey my parents by not observing supererogatory fasts, even though I am able to do so?

Your fast is valid, Allah willing, as long as you ate while not certain of the dawn's breaking and you were unaware of the muezzin's situation. So, if you drank or ate at the time of the adhan (definition), it does not harm your fast, because you were not aware of the dawn's breaking. However, it is more cautious to make up that day if it was an obligatory fast. As for obeying your parents in abandoning a supererogatory fast, you must obey them.