Terms from current post custom taxonomy — excluding parent and only from a single parent term

Each term has a term_id, and each has a parent. Child terms of ParentTerm A will have a parent value equal to ParentTerm A‘s term_id, so given the parent term’s ID, you can test each term to see if the parent field matches-

$parent_term_id = 42;    
foreach ( $terms as $term ) {
    if( $term->parent == $parent_term_id )
        $listing_filters[] = $term->name;
}