Get Parent Custom Taxonomy Term and Color Div background

You’ll have to adjust the markup to suit your specific needs, but the easiest solution would use a post_class filter on the post_class() template tag, to output appropriate classes.

For example, in your template, you need to call post_class(), such as:

<div <?php post_class(); ?>>
<div class="car_type">Ford</div> (background green)
Post Title<br> 
Post Content<br>
</div>

Then, filter post_class, using your term:

function wpse123081_add_post_classes( $classes, $class, $postid ) {

    // Custom Tax
    if ( 'carsCPT' == get_post_type( $postid ) ) {
        foreach ( (array) get_the_terms( $postid, 'carsTax' ) as $term ) {
            if ( empty( $term->slug ) )
                continue;
            $classes[] = 'term-' . sanitize_html_class( $term->slug, $term->term_id );
        }
    }
}
add_filter( 'post_class', 'wpse123081_add_post_classes', 10, 3 );

This will turn this:

<div <?php post_class(); ?>>

…into this:

<div class="term-carTaxTerm">

(along with other classes, of course)

And you can target your CSS accordingly.