Trying to access and print the values from the fields within Paragraph
Here's what worked for me to manually get parts of a field in a paragraph template.

This is in a field template for the actual paragraph, so for me field--field-paragraph.html.twig. I have an image field with the machine name field_image. This prints just the alt text of that image. 

{% for item in items %}
  {{ item.content['#paragraph'].field_image.0.alt }}
{% endfor %}


I looked here for working with paragraph templates: https://www.drupal.org/project/paragraphs/issues/2662766#comment-12528934 and here for general drupal 8 field-getting: https://blog.usejournal.com/getting-drupal-8-field-values-in-twig-22b80cb609bd