How to get only child terms from a custom taxonomy of current post type?

Each term object has a parent attribute which is set to 0 in case it’s a parent root term or the ID of the parent in case it is a child term. So, if you have only one level of child terms then you can check for that attribute to not be equal to 0 in which case the term has a parent.

$object_terms = wp_get_object_terms( $post->ID, 'types', array( 'fields' => 'all' ) );

if ( $object_terms ) {
    echo '' . '' . '' ; /* This line is completely useless. */
    $res="";

    foreach ( $object_terms as $term ) {
        /* If parent would be 0 then this 'if' would evaluate to false */
        if ( $term->parent ) { 
            $res .=  $term->name . ','; /* You probably wanted ', ' here. */ 
        }
    }
    /* This is great. In this form the 'rtrim' is useless. 
       The two "concatenations" are null and completely useless.*/
    echo rtrim($res,' ,').'' . '';  
}

Extra:

function wt_get_child_terms( $post_id, $taxonomy = 'category', $args = array() ) {
    $object_terms = wp_get_object_terms( $post_id, $taxonomy, $args );
    $res="";

    if ( $object_terms ) {      
        foreach ( $object_terms as $term ) {
            if ( $term->parent ) {
                $res .=  $term->name . ', ';
            }
        }
    }

    return apply_filters( 'wt_get_child_terms', rtrim( $res, ', ' ) );
}

echo wt_get_child_terms( $post->ID, 'types', array( 'fields' => 'all' ) );