my_module.module

<?php

/**
 * Implements hook_preprocess_HOOK().
 *
 * Determines the image style to use according to the paragraph parent.
 */
function my_module_preprocess_paragraph(&$variables) {
  /** @var \Drupal\paragraphs\Entity\Paragraph $paragraph */
  $paragraph = $variables['elements']['#paragraph'];

  if (
    $paragraph->bundle() !== 'media'
    || empty($paragraph->field_paragraph_media->entity)
    || $paragraph->field_paragraph_media->entity->bundle() !== 'media_image'
  ) {
    return;
  }

  /** @var \Drupal\Core\Entity\ContentEntityBase $parent */
  $parent = $paragraph->getParentEntity();

  // Default image style.
  $respImageStyle = 'edit_content_full';

  if ($parent->getEntityTypeId() === 'paragraph') {
    // 3 columns paragraphs has even columns so we just need one style.
    if ($parent->bundle() == 'three_columns') {
      $respImageStyle = 'edit_content_third';
    }
    elseif ($parent->bundle() == 'two_columns') {
      // 2 columns paragraphs with 50/50 settings have even columns too.
      if ($parent->field_column_size->value == 'half') {
        $respImageStyle = 'edit_content_half';
      }
      // For 33/66 and 66/33 settings we need to find in which column the paragraph 
      // has been placed.
      else {
        // The field_elements_double field can only contain one paragraph so if it's
        // not ours, it's in the other one.
        $firstColumn = $parent->field_elements_double->target_id == $paragraph->id();

        if ($firstColumn xor $parent->field_column_size->value == 'last-third') {
          $respImageStyle = 'edit_content_third';
        }
        else {
          $respImageStyle = 'edit_content_2thirds';
        }
      }
    }
  }

  // Set the image style name to the entity carrying the field so it can be handled by
  // \Drupal\my_module\Plugin\Field\FieldFormatter\ResponsiveImageFormatter.
  $paragraph->field_paragraph_media->entity->forced_image_style = $respImageStyle;
}

ResponsiveImageFormatter.php

<?php

namespace Drupal\my_module\Plugin\Field\FieldFormatter;

use Drupal\Core\Field\FieldItemListInterface;
use Drupal\Core\Plugin\ContainerFactoryPluginInterface;
use Drupal\responsive_image\Plugin\Field\FieldFormatter\ResponsiveImageFormatter as ResponsiveImageFormatterCore;

/**
 * Responsive image formatter that allow to force the format.
 *
 * If the entity that carries the image has a "forced_image_style" value
 * this is the one used in place of the image style selected in the formatter
 * configuration.
 *
 * @see my_module_paragraph_preprocess_paragraph().
 *
 * @FieldFormatter(
 *   id = "my_module_responsive_image",
 *   label = @Translation("Responsive image (forced by parent)"),
 *   field_types = {
 *     "image",
 *   },
 *   quickedit = {
 *     "editor" = "image"
 *   }
 * )
 */
class ResponsiveImageFormatter extends ResponsiveImageFormatterCore implements ContainerFactoryPluginInterface {

  /**
   * {@inheritdoc}
   */
  public function viewElements(FieldItemListInterface $items, $langcode) {
    $entity = $items->getEntity();
    if (!empty($entity->forced_image_style)) {
      $this->setSetting('responsive_image_style', $entity->forced_image_style);
    }

    return parent::viewElements($items, $langcode);
  }

}