Is it permissible to fast the six days of Shawwal despite the parents' objection, out of fear for their daughter's health, knowing that she has completed her make-up fasts and wishes to earn reward?

If you intend to fast the six days of Shawwal, seek your parents' permission first. If they permit you, then fast. If they prevent you for a valid reason, such as your weakness to undertake it, then obedience to them is obligatory. However, if the prevention is merely out of compassion, then their obedience is not binding.