How to use parent paragraph or field name in template_preprocess_file_link function?

Question

We created custom theme suggestion for fields links using this code

function THEME_theme_suggestions_file_link_alter(array &$suggestions, array $variables) {
  $route = \Drupal::request()->attributes->get('_route');
  // Adds a template suggestion for all files displayed as links, except in node/add and edit forms
  if ( strpos($route, 'add') === FALSE && strpos($route, 'form') === FALSE ) {
    $custom_suggestions[] = 'file_link__button';
  }
}

This allows us to use template_preprocess_file_link() to add more attributes to the file (MIME, weight, description...) like this :

/**
 * Prepares variables to recreate link in file-link--button.html.twig
 * file name & uri, formatted type (with user friendly names) & weight
 *
 */
function THEME_preprocess_file_link__button(&$variables, $hook) {
  // Define user-friendly MIME types name
  $mimes_types_name = [
    'application/vnd.ms-excel' => 'Excel',
    'application/pdf'          => 'PDF',
    // ...
  ];
  $file = $variables['file'];
  $file_type = $file->getMimeType();
  $variables['file_type'] = $file_type;
  if (isset($mimes_types_name[$file_type])) {
    $variables['file_type_formatted'] = $mimes_types_name[$file_type];
  }
  // ...
}

However in none of those function we can access the field name or better its parent to make a dedicated Twig template depending on paragraph parent or field machine name.
In THEME_theme_suggestions_file_link_alter() we could targetthe node type (using the route) but we're using paragraphs and we can't access the parent paragraph.

Any suggestion to make more than one Twig template for template_preprocess_file_link() ?