Is it permissible for me to take the financial difference for myself when exchanging currency at the black market rate, while giving my sister the amount at the bank rate, knowing that my father sends her allowance to my card after her card was lost?

It is impermissible to take any increase from the money entrusted to an agent without the principal's permission, because such increase reverts to the principal. The evidence for this is that the Prophet (peace be upon him) gave Urwah ibn Al-Ja'd a dinar to buy a sheep. He bought two sheep, sold one for a dinar, and returned to the Prophet (peace be upon him) with a dinar and a sheep.