Preface: Do not use that method from that blog, there is a better method.
something._referringItem is not cache safe, so better not use this inside Twig templates.
something._referringItem doesn't perform any access checks
paragraph._referringItem.parent.parent.entity is possibly the longest and most confusing alternative to write paragraph.parentEntity
So, now the real answer :-)
paragraph.parentEntity is returning the raw parent entity object (very similar to the paragraph variable) but it does not include a render array like content. So you neither need .content nore you need |field_value.
Instead head directly for the field value parent.field_on_hostnode.0.value
More info about variables in entity templates:
content = complete render array of an entity
content.field_something = render array of a single field
content.field_something|field_value = get raw field value back from a render array (requires "Twig field value" module)
entity = raw entity object (replace entity with node in node template, paragraph in paragraph template, etc).
entity.field_something.value = not recommended, will behave differently depending on field type and field cardinality
entity.field_something.getValue() = get an array of raw field values, works for all field types and all field cardinalities
entity.field_something.0.value = get the first raw data field value (for plaintext fields, numbers, emails, telephone field; will not work with links, images, entity references...)
entity.field_something.0.name_of_database_column = get the first raw data subvalue name_of_database_column of field_something (this could be format in a formatted text field, end_value in a daterange field, target_id in a entity reference field,...)
Entity reference field only:
Note that the following methods are not cache-safe, not language-aware and do not perform access checks.
entity.field_reference.0.target_id = get the numeric ID of the first referenced child entity
entity.field_reference.0.entity = the first referenced child entity object (no render array!)
entity.field_reference.0.entity.getTranslation('de') = get the german translation of the first child object
Paragraphs only:
paragraph.parentEntity = the direct parent entity object (no render array!)