How to add class to embedded Paragraphs field-items?
Drupal 7
You should be able to copy this function into your template.php file of your theme:
/**
 * Process variables for paragraphs-items.tpl.php
 */
function mytheme_preprocess_paragraphs_items(&amp;$variables, $hook) {
  $variables['view_mode'] = $variables['element']['#view_mode'];
  $variables['field_name'] = $variables['element']['#field_name'];

  $variables['content'] = $variables['element']['#children'];

  $variables['classes_array'][] = drupal_html_class('paragraphs-items-' . $variables['element']['#field_name']);
  $variables['classes_array'][] = drupal_html_class('paragraphs-items-' . $variables['element']['#field_name'] . '-' . $variables['view_mode']);
  $variables['classes_array'][] = drupal_html_class('paragraphs-items-' . $variables['view_mode']);

  $variables['theme_hook_suggestions'][] = 'paragraphs_items__' . $variables['element']['#field_name'];
  $variables['theme_hook_suggestions'][] = 'paragraphs_items__' . $variables['element']['#field_name'] . '__' . $variables['view_mode'];

  // your code here 
  $variables['classes_array'][] = 'your-class';
  $variables['classes_array'][] = 'your-other-class';
}

You should also be able to do this for individual item:
/**
 * Process variables for paragraphs-item.tpl.php
 */
function mytheme_preprocess_paragraphs_item(&amp;$variables, $hook) {
  // your code here 
  $variables['classes_array'][] = 'your-class';
  $variables['classes_array'][] = 'your-other-class';
}

After adding this function, clear the cache so it is registered. You can then check if either one are being called by adding a breakpoint or print 'called'; exit; if you do not have something like XDebug installed.
Note how the second function is paragraphs_item and not paragraphs_items. Both of these are defined as theme hooks in paragraphs.module.
You can even do this as well:
/**
 * Implements hook_preprocess_entity().
 * @param $variables
 */
function mytheme_preprocess_entity(&amp;$variables) {
  if ($variables['entity_type'] == 'paragraphs_item' &amp;&amp; $variables['elements']['#bundle'] == 'your_bundle') {
    $variables['classes_array'][] = 'your-class';
  }
}