Have you heard of SWOT Analysis? May be that works for you to put your case forward to your father. SWOT Analysis is nothing but writing down your:

Strengths: Why you think you are ready for JEE Mains in January

Weaknesses: Any weaknesses that your father sees in you that he thinks you are not ready

Opportunities: Any areas of improvement that is required in your subjects and that can help you appear for January session

Threats: What is stopping your father from allowing you to write JEE Mains in January

Make a sheet of this analysis and discuss this with your father. If you have already completed your 12th, reappearing for your 12th means you will be facing tough competition from your juniors when you have to appear for JEE Mains again. Discuss this with him and do explain to him why you think you are prepared.