Rebekka, you do need to inform your parents about reverting to Islam at the moment if you do not feel comfortable, nor do you require the permission of anybody to become a Muslim. If you believe Islam to be the religion of truth, then try to become a Muslim as soon as possible and you can inform your parents later on down the road whenever you feel comfortable.
We do not know your parents, so we do not know what their understanding of Islam is. You are their daughter, you would know what questions and concerns they may have about Islam. It would be best for you to become a Muslim, gain more knowledge, so that when you do feel comfortable, you would be able to not only tell them of your decision but also answer their questions and concerns that they may have about Islam.