How can I convince my parents to let me go on a summer abroad program?

If you read the papers today about Britannee Drexel (also a good student, etc) you would understand why your parents don't want you to go. I agree it's not safe for a 16 year old to go to Madrid without parents. It's not about you being irresponsible or deficient in any way. It's about the world we live in and the reality that bad things do happen. Give your parents a break and wait a couple of years.