s others have pointed out, there are a lot of really good music schools out there, others not mentioned include CIM (Cleveland Institute of Music), U Michigan, Northwestern, USC, UCLA. There also is Colburn in LA, that is very tough to get into, but is also totally free ride (tuition, room and board).
Juilliard is a great school, but on top of the 6-7% acceptance rate, also know that it is especially difficult on piano, they are flooded with applicants from around the world, especially from Korea and China, so it may even be tougher than that number indicates. However, the real answer is you never know, the audition process at Juilliard is not scientific, and what they look for on a pre screen or in an audition is not scientific, so you truly don’t know. So you apply for the pre screen and you don’t get an audition date, so what? Unless the money is that big an issue for the application (and it can be for some people), at least you tried. And if you are seriously interested in music, as another poster pointed out, in any kind of music you will face this situation, especially as a pianist. Orchestral instrument musicians if they audition for any kind of decent orchestra, might be competing against 150 other people, and if someone looks and says “I’ll never get in there, so why audition?”, they will end up not doing anything…
My son auditioned at a school he was pretty certain he would get into, but also put down a teacher he thought he had little chance of getting into the studio io, this guy takes very few students each year and most of them are grad level, and was shocked when he got in. If he looked at who was getting into the studio, the kind of kids that often have artists representation and have won major competitions, it would be very easy to say “I don’t stand a chance in h*ll” of getting in, but he did, even though he was none of those when he got in.
Why did your parents tell you not to apply to Bard? Bard requires a dual degree from its students, so while studying music you would also be getting another degree, maybe that would influence your parents, if they think getting a conservatory degree is ‘not real’ or something…