Add parent ID to body_class

Using the body_class filter

// Use the filter 'body_class'
add_filter( 'body_class', 'parent_id_body_class' );
function parent_id_body_class( $classes ) {
    // add parent id number to the $classes array
    $classes[] = wp_get_post_parent_id( $post_ID );
    // return the $classes array
    return $classes;
}

Anyway, I recommend you to add some text to the id number, in order to make it more comprehensive, as follows:

// Use the filter 'body_class'
add_filter( 'body_class', 'parent_id_body_class' );
function parent_id_body_class( $classes ) {
    // add comprehensive text followed by parent id number to the $classes array
    $classes[] = 'parent-id-' . wp_get_post_parent_id( $post_ID );
    // return the $classes array
    return $classes;
}

I hope it helps.