Your parents are right. Study abroad. (Especially if in Switzerland !)
My first impression is that the schools might be the University of Denver versus Trinity College in Dublin, Ireland or St. Andrews in Scotland.
OP: Always be honest with your schools. No need to be concealing your plans as there are plenty of US schools from which to chose.
Attending school abroad for a year will make you a more attractive applicant to US schools.
Important to study abroad when young & developing.