What is the ruling on not going to the mosque out of shyness from neighbors asking about one's father?

Performing obligatory prayers in congregation is a duty upon the mature Muslim, unless there is an excuse, as the Prophet, peace and blessings be upon him, said: "By Him in Whose Hand my soul is, I was about to order that firewood be collected, then I would order the prayer to be called, then I would order a man to lead the people in prayer, then I would go to some men and burn their houses down upon them." Allah Almighty commanded congregational prayer even in times of fear and travel, so it is even more incumbent in times of security and residency. You must perform prayer with Muslims in the mosques, and gently advise your father to join you in the mosque.