WordPress – registering sidebar and adding a button directly after .textwidget

Well as long as the buttons are all the same:

if (function_exists('register_sidebar')) {
 register_sidebar(array(
    'name'=> 'Parallax Area',
    'id' => 'parallax',
    'before_widget' => '<li class="parallax-box">',
    'after_widget' => '<button class="btn-action">Learn More...</button></li>',
    'before_title' => '<h2>',
    'after_title' => '</h2>',
));
}

If the buttons were not all the same, you could make them be the same and specify the target href in the .textwidget on the button click action with jQuery:

 <li class="parallax-box">
            <h2>Computer Repair</h2>            
            <div class="textwidget" data-href="http://link/to/your/content">Here will be some text that explains the contents of this box and allows them to click through to a page.

            </div>
           <button class="btn-action">Learn More...</button>
 </li>  

.

    <script type="text/javascript">
        jQuery(document).ready(function() { 
            (function ($) { 

                $('.btn-action').on('click',function(e){
                    e.preventDefault();
                    var url = $(this).sibling('.textwidget')[0].getAttribute('data-href');
                    window.location.href=url;
                });

            })(jQuery);
        });
    </script>