Should I direct the college costs to my parents’ account or mine?

College costs are always billed to you (the student) since they are your charges. The school doesn’t care how they get paid or by who, just as long as they get paid. If your parents are helping pay your educational costs work out a way where they will get a copy of the billing statement (you make a copy and send it to them, they can access it through the school’s website, etc.). Then they can write the check and send it in.