How to render inside a paragraph template a specific view mode for that paragragh You may say that I should just change the display settings but in my use case I need to include 2 view modes inside one template as the mobile and desktop version of the output require very different HTML.

Is there a way to do this or do I require some preprocess function to do this?

Thanks
You could try using Twig Tweak module 

{{ drupal_entity('paragraph', NUMERIC_ID, VIEWMODE_MACHINE_NAME) }}

Also see the cheat sheet  for Twig tweak, it really is a great module for frontend work!