How to get parent and child name in taxonomy page

You can use get_ancestors() to get an array of IDs of the ancestors of a term in any taxonomies, and in your case, you would use the last 2 items like the example below, where I used:

• get_queried_object_id() to get the ID of the term on the term archive page that you are currently viewing
• get_term_field() to get the term’s name, but you can use get_term() instead
• get_term_link() to get the URL of the term’s archive page

// Get the current term ID.
$cat_id = get_queried_object_id();

// Get the term's ancestors.
$parents = get_ancestors( $cat_id, 'car-brand' );

// Get the level 1 ancestor term, and display it.
$brand_cat_id = array_pop( $parents );
printf(
    'brand: <a href="%s">%s</a><br />',
    esc_url( get_term_link( $brand_cat_id ) ),
    esc_html( get_term_field( 'name', $brand_cat_id ) )
);

// Get the level 2 ancestor term, and display it.
$series_cat_id = array_pop( $parents );
printf(
    'series: <a href="%s">%s</a>',
    esc_url( get_term_link( $series_cat_id ) ),
    esc_html( get_term_field( 'name', $series_cat_id ) )
);