score:9
The most likely reason the passport office would care about your girlfriend's parents' nationality would be in case it is necessary to determine whether she is a US citizen. I assume that she is a US citizen independent of her father's nationality (which would only be relevant if she had been born outside the US, and, depending on when she was born, on her mother's circumstances). If that assumption is correct, then she should put "unknown" for her father's citizenship.