I can’t change the body background-color through style.css

by

It sounds like you just need to remove support for the custom background color.

remove_theme_support( 'custom-background' );

You will probably need to hook it to get it to run after the parent functions.php

function disable_bg_wpse_97248() {
    remove_theme_support( 'custom-background' );
}
add_action('after_setup_theme','disable_bg_wpse_97248',100)