Is the daughter exempt from paying her father's phone bill if she was unaware of the amount due beforehand, given that she is unemployed and receives a small monthly allowance from her father?

Ignorance does not negate liability for damages. The majority of imams and scholars hold that liability for damages, blood money (diyat), and the rights of individuals are never waived. Error and intent are the same when it comes to people's wealth. Based on this, your daughter's ignorance of the call charges does not exempt her from liability for the resulting amount. Her belief that the cost of calls is your obligation is incorrect, as it is not part of the obligatory maintenance (nafaqah) which includes bread, stew, clothing, housing, and their accessories.