Instead of trying to convince my parents why BS offers a superior education in general, I focused my efforts on convincing them why BS offered a superior education in the context of my individual interests and aspirations (clubs, sports teams, organizations etc).
I respectfully disagree with @Imartzi’s reasoning. If you are applying to boarding school just to put it on a college app or get ahead in the admissions process, your thought process is flawed. Also, you cannot assume that BS has “better” classes or teachers than the LPS. There is a broad range of boarding schools and teachers within said institutions.
I hope my comment provided some insight. If you have any remaining questions feel free to DM me. Good luck!