Accessing Webforms Parameters in hook_form_alter
I can't see a way to access the originalData property you mention. It's a protected property and there's no getter defined. See the callback_object's class in the webform module at src/WebformSubmissionForm.php.
I'd take a slightly different approach here. Since you are building custom values based on the paragraph id, it seems appropriate for the paragraph id to be included with the submission data. To do this, add a new hidden element to your webform for the paragraph id, e.g. with machine name paragraph_context.
Then, to provide a default value for your hidden field, update your twig like:
{{ { 
  '#type': 'webform',
  '#webform': 'contact', 
  '#default_data': {
    'paragraph_context': paragraph.id,
  },
} }}

(Note: I don't see any mention of the values or parameters options you're using in the How to embed a Webform documentation on d.o.)
Lastly, access your paragraph_context value via the #default_value key for your hidden element in your hook_form_alter(). For example:
mymodule_hook_form_alter(array &amp;$form, FormStateInterface $form_state, $form_id) {
  if ($form_id == 'my_form_id' &amp;&amp; !empty($form['elements']['paragraph_context'])) {
    $paragraph_context = $form['elements']['paragraph_context']['#default_value'];
  }
}