How can I prevent cache on any page where a paragraph appears? I am using the paragraphs module with a custom theme I created. I want to figure out how to clear the cache for any page that a specific paragraph appears on. The custom theme is composed of a bunch of templates (i.e. paragraph--paragraph-name.html.twig) and a *.theme file where I preprocess each of the paragraph types.
THEME_preprocess_paragraph__paragraph_name

I don't want to disable the entire site's cache. Can I clear the cache in this file? In this way somehow?
You probably want to be really heavy handed and use \Drupal::service('page_cache_kill_switch')-&gt;trigger();
If you put that in a custom module in hook_entity_view_alter() then you can add a conditional that fires it when your paragraph entity is being built.