Praise be to Allah, the Lord of the Worlds; and blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
If your father had died after the Zakah had become obligatory on his wealth, then you would have been obliged to pay its Zakah and distribute it among the needy people. And it should be given before distributing inheritance among his heirs. It is like a debt, which is claimed. Moreover, this Zakah became the other's right after it had become due.
And you are not obliged to pay Zakah on your property (inherited or not) unless one Islamic year (Lunar) has passed. Muslim scholars support this viewpoint.
And Allah knows best.