get post attachments of a particular size

The image attachment type is stored in the postmeta table in the _wp_attachment_metadata meta key. The data is serialized, so it can’t be filtered on. But you could use a meta_query in your query:

$images = get_posts( array(
  'post_type' => 'attachment',
  'orderby' => 'rand',
  'posts_per_page' => -1,
  'meta_query' => array(
        array(
            'key' => '_wp_attachment_metadata',
            'value' => 'extra_large',
            'compare' => 'LIKE'
        )
    )
) );

foreach ($images as $img) :
  $image = wp_get_attachment_image_src( $img->ID, 'extra_large' );
  if ( !empty($image) )
    $carousel[] = array($img->post_parent, $image);
endforeach;

That would be much more efficient — because it should only retrieve the attachments that have the ‘extra_large’ version.

Alternative Option:
Attach the images to a post, and then just use post_parent in your query, as is shown in the answer of this question: Get all image from single page