Full needs schools typically determine your need based on their own system and include additional information that may not be part of the fafsa application. This means their idea of what you need may be really different from the FAFSA EFC or what you personally feel you need.
I can only speak for our own experience but having a straight forward financial situation, the full need schools our two were accepted into did not include parent loans. Some other private schools that DONT meet need did. Neither applied to USC though.