I am 16 and am on my father's insurance, what are some options for me to be placed on birth control without him knowing?

Adolescent medicine. This is a specialty which is most helpful with questions like yours. This is a pediatrician with 3 more years of training with a special interest in people your age. S/he can help. You may seek a public clinic or pay for birth control yourself. You are doing the right thing in seeking the available options in protecting yourself against unwanted pregnancy. Well done.