Just to be clear (and perhaps harsh), your attitude is very self-centered and a poster-child for the definition of “Entitled.” While many do not complain when they have to find a corner to study after coming home from a poor inner-city school, and have to justify (to their parents/family) taking time for homework instead of helping out with home chores or the home business; you are “mad and bitter” that you may be forced to go to your second-choice elite private boarding school.
I certainly hope you were in a different frame of mind when you wrote your essays, because ‘entitlement’ is unbecoming and has a way of seeping through into one’s writing!
And be happy with your parents. They are considering spending discretionary money that they could use on sending themselves on a photo-safari in Africa, or heli-skiing in Canada, or myriad other choices, on educating you, when perfectly good free public options are available. Anything beyond LPS is a gift. Don’t be “mad and bitter” over a gift, even if the giver is trying to control what is given.
Lecture over.
Control what you can control. One thing you can control is convincing your parents of why Hotchkiss and St. Paul’s are better for you (as recommended by other posters). Another is to make sure your sister’s favorites are aligned with yours, so if you both get in to all of the schools, her top choice will be the same as yours.