Is it obligatory to obey one's parents if they forbid me from going to the mosque and threaten to expel me from the home?

Congregational prayer is obligatory unless there is a valid excuse. A father preventing his son from congregational prayer is not permissible. Try to gently convince your parents of the obligation of congregational prayer. If they permit you, pray in the mosque. If they do not permit you and you fear being expelled, then pray at home with a congregation if possible, in accordance with the verse: "So fear Allah as much as you are able."