Parents may know what is a “better” school, but the student knows better about “fit”. Being in the right environment is a big part of being successful in college.
If you really want school X, then convince them. Give them all the reasons why you think X is better for you. You have to take what they said into account, but they will take what you said into account too. “Who has more say” is the wrong way to go about it. If you all talk it out with detailed reasons, I am sure you guys will come to an agreement. And even if you didn’t solidly, one side will feel better about the choice being made even if it wasn’t their initial choice.
But sounded like you have more schools to hear from, so don’t worry about this just yet, but start thinking about pros/cons of X or Y, and maybe visit them to get a better feel, if possible.
Good luck!