How to retrieve value of hidden field in twig?
The variables in a paragraph template contain the fully loaded paragraph entity:

paragraph.html.twig

{#
/**
 * @file
 * Default theme implementation to display a paragraph.
 *
 * Available variables:
 * - paragraph: Full paragraph entity.


from which you can retrieve raw values of any field, not matter whether base field or configured field or displayed or not:

{% set value = paragraph.field_example.value %}


On the contrary {{ content }}

 * - content: All paragraph items. Use {{ content }} to print them all,
 *   or print a subset such as {{ content.field_example }}. Use
 *   {{ content|without('field_example') }} to temporarily suppress the printing
 *   of a given child element.


contains only the fields configured to be displayed in "Manage Display".