Is it permissible for me to slaughter an 'aqīqah for my brother or sister out of my own money, knowing that my father is unable to do so, and I have already performed 'aqīqah for myself?

The 'Aqiqah is a confirmed Sunnah for the guardian of the newborn who is able to perform it. It is best to perform it on the seventh day; otherwise, on the fourteenth; and if not, then on the twenty-first. If it is delayed, it is sunnah for the guardian to perform the 'Aqiqah at any time until the child reaches maturity. Some scholars have deemed it desirable for the child, once they reach maturity and no 'Aqiqah was performed for them, to perform it for themselves.

Based on this, the default is for the father to perform the 'Aqiqah for his children. If he does not do so, there is no impediment for another person to perform the 'Aqiqah for them as a charitable act and kindness. However, it is advisable to seek the father's permission for this, to avoid any difference of opinion.