A member of my family informed me that one has to pay Zakah al-Fitr not only for his family members but also for all those who have invited them in Ramadan.

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Zakat-ul-Fitr is obligatory upon every Muslim who owns extra provisions beyond his need as well as the need of everyone in his family on the day and night of Id-ul-Fitr. In other words, one (father...) should pay for himself and for whom he supports.
For sure, none among Muslim scholars believes that Zakat-ul-Fitr should be paid on behalf of those he invite them for food in Ramadan. This statement is strictly rejected without any form of difference among the Muslim Ummah.
Allah knows best.