Conclusion: Both your father and your mother must sacrifice a hadiy for their tamattu‘. You do not have to offer the sacrifice on their behalf; rather if they cannot afford a sacrifice, then they have to fast three days during Hajj and seven days when they return home. They can start to fast this three days from the time when they enter ihram for ‘umrah. And if they are not able to fast, then they do not have to do anything. As for you, it is permissible for you to enter ihram for Hajj ifraad [Hajj on its own] when you set out for Makkah, and you do not have to offer a hadiy, because even though you intended to do Hajj in the same year, the fact that you travelled to your city of residence, namely Riyadh, cancelled out your tamattu‘. But you should enter ihram for Hajj before reaching the miqaat, and not pass the miqaat except in a state of ihram. When you have entered ihram for ifraad, it is not permissible for you to exit from it or to wear ordinary clothes; rather you must remain in your ihram until the pilgrims exit ihram on the day of sacrifice (yawm an-nahr). And Allah knows best.

There is nothing wrong with doing ‘Umrah on the Day of ‘Arafah or the Day of ‘Eid because ‘Umrah can be done at all times.