Will the questioner be absolved of responsibility by taking her mother's full salary and giving it to her father with the intention of it being money she previously took from him, knowing that her mother gives her father her salary as assistance, and without informing the father?

Your responsibility is discharged by paying the father what the mother gave you. The fact that the mother used to give her salary to her husband as a donation does not affect this, because what she gave you is not the father's property; rather, it remains the mother's property. Therefore, there is no sin upon you in paying it to your father as his right, and you are not obliged to inform him that it is a repayment of his right.