My theory: Red and Mary-Louis Parker had a (ballet-dancing daughter) before they divorced. Then, Mary-Loise had a 2nd daughter, Elizabeth, with a 2nd husband/boyfriend. Red's enemies tried to hurt him by destroying his family, setting the fire that killed his daughter and her step-father, Elizabeth's biological dad. Elizabeth's mom, Mary-Louise Parker, wasn't home at the time of the fire. Red saved Elizabeth, the younger sister of his dead daughter, and gave her to her adoptive father for her own safety. He did this on the condition that her mother, Mary-Louise Parker, never try to find her. Mary-Louise Parker took on a new identity/went into Witness Protection and remarried. For the last 20 years, she has kept her promise not to seek out Elizabeth. Red had her blindfolded when handed over by Berlin so that she would not see and potentially recognize her daughter, Elizabeth, thus revealing Red's culpability in the destruction of Elizabeth's biological family. Red feels fatherly towards Elizabeth because she is the half-sister of his dead daughter: dead as a result of his own deeds.