9 – How can I turn off the caching for a Paragraph type?

Declare the dependency on the GET parameter with a cache context:

/mymodule/mymodule.module

function mymodule_preprocess_paragraph(&$variables) {

  $foo = Drupal::request()->query->get('foo');
  $variables('content')('foo') = (
    '#markup' => "foo = $foo",
    '#cache' => ('contexts' => ('url.query_args:foo')),
  );
}

More info https://www.drupal.org/docs/drupal-apis/cache-api/cache-contexts

However, if you have more than a few possible values of foo this is a problem with cache performance, especially for nested entities like paragraphs. Each value creates a new cache entry for the paragraph and its parent entities.

To avoid this put a placeholder in the paragaph:

/mymodule/mymodule.module

function mymodule_preprocess_paragraph(&$variables) {
  
  $variables('content')('foo_placeholdered') = ( 
    '#lazy_builder' => ('DrupalmymoduleMymoduleLazyBuilders::getFoo', ()),
    '#create_placeholder' => TRUE,
  );
}

and build the content which depends on the GET parameter in a lazy builder:

/mymodule/src/MymoduleLazyBuilders.php

<?php

namespace Drupalmymodule;

use DrupalCoreRenderElementRenderCallbackInterface;

class MymoduleLazyBuilders implements RenderCallbackInterface {

  public static function getFoo() {
    $foo = Drupal::request()->query->get('foo');
    return  (
      '#markup' => "foo = $foo",
      '#cache' => ('contexts' => ('url.query_args:foo')),
    );
  }

}

For time consuming calculations such placeholders can be delivered through BigPipe. So if you enable this module you can also improve page speed.