Wordpress - Check if is on child-page of a particular page

You can do that with $post->post_parent. You will have to check if child page's parent is Services page. So this is how you will check it.

I assumed 123 in following code is page ID of your services page. Replace it with actual ID.

if ( 123 == $post->post_parent ) { ?>
    <div class="col-md-2 col-sm-4">
        <?php ci_e_logo('<h1 class="logo ' . get_logo_class() . '">', '</h1>'); ?>
    </div>
<?php }

You can get the Post page / post name using this method.

$parent = array_reverse(get_post_ancestors($post->ID));
$page_parent = get_page($parent[0]);
echo $page_parent->post_name;

you can use the condition as per your requirements.

<?php 
global $post;

if ( is_page('home') || is_page('services') ) { ?>
    <div class="col-md-2 col-sm-4">
        <?php ci_e_logo('<h1 class="logo ' . get_logo_class() . '">', '</h1>'); ?>
    </div>
<?php } 

elseif ( preg_match( '#^service(/.+)?$#', $wp->request ) ) { ?>
    <div class="col-md-2 col-sm-4">
        <?php ci_e_logo('<h1 class="logo ' . get_logo_class() . '">', '</h1>'); ?>
    </div>
<?php
}

else { ?>
    <div class="col-md-2 col-sm-4">
        <h1 class="logo imglogo">
            <a href="<?php echo esc_url( home_url( '/' ) ); ?>">
                <img src="<?php echo get_bloginfo('template_directory');?>/images/picturehere.png" alt=""></a>
            </h1>
    </div>
<?php } ?>