Getting a custom block image field URL in the block Twig template
#url at this stage is not processed yet. This happens later in the image templates.

To get the url in the block template you can access the uri field in the image file entity instead:

{{ file_url(content.field_parallax_image.0['#item'].entity.uri.value) }}


BTW you didn't find this property in the dump output, because it is a computed value to make it easier to handle entity references.