Is it permissible to disobey one's parents and quit one's job because one feels a sense of unworthiness for responsibility? And is the resulting earning considered unlawful? And does charity purify it if it is mixed with the unlawful?

If you are truly unable to perform the work as required, then it is not permissible for you to remain in it unless you inform the employer of the shortcomings and they grant you permission to stay. If they do not grant permission, then it is not permissible for you to stay through deception, and the salary is not lawful for you if you do not perform the agreed-upon work. Giving some of it as charity does not absolve you, because you do not deserve it. And there is no obedience to your parents in remaining in this work in such a manner, for the Prophet, peace and blessings be upon him, said: "Obedience is only in what is good and right," and there is no obedience to a creature in disobedience to the Creator. The matter depends on the school administration's decision after you inform them.