The Cullens are a family. They have a special connection thanks to Carlisle. He was kind of the creator of everyone. He thought he would never have children until he met you.
It was actually Edward who saved you at the age of ten when he found you in the middle of the night bleeding and almost fainting on the streets. What he could hear through your cries was that you had just escape your abusive parents (that until now Edward would have been ready to bite off and cut their throats for what they did to you). Your strength was actually what gave him strength and saved you without killing you, and by that time he immediately adopted you; that time Edward hadn’t manage his thirst like nowadays. He saw you as something more than a child...his own baby, child.
He was your father. He showed you how to walk, sit, act after the incident from your abusive parents, like a baby. He comforted you,he helped you, and feed you, he did everything for you, treating you like a defenseless fawn.
It was your first few days, once Edward saved you and brought you to the Cullen mansion, in some ways he was your savior, but also most importantly your father. Edward's golden eyes watched you closely as a father watched and controlled how his baby took the first steps towards him.