Conclusion: The person who does tawaf and sa‘i should do them walking, unless he has an excuse. That is preferable according to scholarly consensus. If he does tawaf or sa‘i whilst riding with no excuse, then we have explained above that there is a difference of scholarly opinion concerning that. What appears to be the case is that his tawaf and sa‘i are valid, in sha Allah. However, it is not appropriate for a person to do that in the first place, so as to be on the safe side with regard to his worship, and so as to avoid a matter concerning which the scholars differed. Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: Some people are careless regarding that, and they do sa‘i in wheelchairs with no excuse, even though many of the scholars have said that doing sa‘i whilst riding is not valid without any excuse. This matter is a matter concerning which the scholars differed, namely: with regard to sa‘i, is it stipulated that the individual must do it walking unless he has an excuse, or is it not? But the individual should be prudent with regard to religious matters, and he should do sa‘i walking so long as he is able to do so, and if he is not able to do that, then Allah does not burden any soul with more than it can bear. End quote from Majmoo‘ al-Fataawa (22/451). And Allah knows best.

These are some of the conditions of tawaf: being Muslim, being of sound mind, forming the intention, covering the `awrah, being in a state of purity, performing seven complete circuits, keeping the Ka`bah on one’s left, performing tawaf on foot, if able to walk, and the circuits must be done consecutively.

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.