I am an Indian passport holder working in Bahrain. I want to visit my father in Riyadh. Is there a one week visa available for such a journey to Saudi Arabia? If not, is there any other option? A Transit visa maybe?
Transit visas are not applicable to you (you would need a permanent resident visa in a bordering country (other than Qatar) to apply for a transit visa), there are also no visas on arrival for Saudi Arabia.
Your only option therefore is to apply for a family visit visa (your father can request it on your behalf). Once approved, approach the embassy to have it stamped on your passport.
It will be a single entry visa valid for 90 days.