Get paragraph id in field template
The clean solution is template preprocessing: In yourtheme.theme create something like

function yourtheme_preprocess_field(&amp;$variables, $hook) {
  $element = $variables['element'];
  $parentEntity = $element['#object'];
  //might be useful for if-clause: $variables['field_name'] and $variables['field_type']
  if ($parentEntity &amp;&amp; $parentEntity-&gt;getEntityTypeId() == 'paragraph' ) {
    //whatever logic you need goes here, e.g. something like
    //$variables['attributes']['data-from-parent-paragraph'] = $parentEntity-&gt;getYourStuff();
  }
}