Is it permissible for me to take the money my father gave me, after I swore by Allah not to take it, and my mother gave it to me to pay off a debt after she herself repaid my debt from the same amount?

The intention of the swearer and the reason for their oath specify and restrict it. If the questioner, by their oath not to take the money, meant specifically the aforementioned situation, then they do not break their oath if they take it again when needed, whether from their father or mother. As for settling the questioner's debt with this amount, it does not fall under the ruling of 'taking' unless they intended absolutely no benefit from it, whether by taking it directly or by using it for a personal benefit. Ibn Qudamah said: "The basis of an oath is the intention of the swearer," and "Reasons are considered in oaths."