Remind your parents of the sacrifice they are making by home schooling you. They sacrifice their time, money and often have little time for a life of their own. Tell them because home schooling has disadvantages for both you and them, you think it's time you go to public school.
Use the excuse of home schooling costing too much. Home schooling can be very expensive and at least one parent usually has to give up their job. Tell your parents you think it would be better if this money went towards your college education.
Explain to your parents the disadvantage home schooling has made to your social development. Advise your parents your need to interact with your peers and to have relationships outside of the family.
Tell your parents you want to play sports. Tell them you think sports would be a good way for you to learn team work, learn discipline and make friends.
Make the argument you need to prepare for life away from home. Tell them you'll be going to college one day and need to get used to being away from home. Explain public school would be a way to gradually get used to the outside world.
Inform your parents you can handle the disadvantages of public school. Tell them while you appreciate the effort they have made by home schooling you, because they have raised you right and taught you the right values, you're now ready to handle public school.