Exclude a category from a query that includes its parent category

First you should move wp_reset_query(); outside of the loop instead of resetting the query every post of the loop.

And as for your query you can use category__not_in

$args = array(
    'cat' =>  14,
    'category__not_in' => array('45'),
    'posts_per_page' => 20
);

$the_query = new WP_Query($args);
while ($the_query->have_posts()) : $the_query->the_post();?>    
            <li>
            <a href="https://wordpress.stackexchange.com/questions/17387/<?php the_permalink() ?>" rel="bookmark" class="postTitleLink">
            <?php the_title(); ?></a>
            </li>
<?php endwhile; ?>
<?php wp_reset_query(); ?>

this will exclude all posts of category 45.