No wait I've solved it:
You have a miss-match between the field in your IF parser and what is displayed
{{#if: {{{relatives<includeonly>|</includeonly>}}} | {{!}} '''Parents:''' {{!!}} style="text-align:left;" {{!}} {{{parents}}} }}
|-
|- style="font-size:2.5; vertical-align:middle; border-bottom: 1px solid #DDA0DD;"
{{#if: {{{relatives<includeonly>|</includeonly>}}} | {{!}} '''Siblings:''' {{!!}} style="text-align:left;" {{!}} {{{siblings}}} }}
|-
|- style="font-size:2.5; vertical-align:middle; border-bottom: 1px solid #DDA0DD;"
{{#if: {{{relatives<includeonly>|</includeonly>}}} | {{!}} '''Aunts and Uncles:''' {{!!}} style="text-align:left;" {{!}} {{{aunts and uncles}}} }}
|-
|- style="font-size:2.5; vertical-align:middle; border-bottom: 1px solid #DDA0DD;"
{{#if: {{{relatives<includeonly>|</includeonly>}}} | {{!}} '''Cousins:''' {{!!}} style="text-align:left;" {{!}} {{{cousins}}} }}
Where you have "relatives" written, it should be "parents" or "siblings" etc, otherwise the parser is waiting on a trigger that is never activated:
{{#if: {{{parents<includeonly>|</includeonly>}}} | {{!}} '''Parents:''' {{!!}} style="text-align:left;" {{!}} {{{parents}}} }}
|-
|- style="font-size:2.5; vertical-align:middle; border-bottom: 1px solid #DDA0DD;"
{{#if: {{{siblings<includeonly>|</includeonly>}}} | {{!}} '''Siblings:''' {{!!}} style="text-align:left;" {{!}} {{{siblings}}} }}
|-
|- style="font-size:2.5; vertical-align:middle; border-bottom: 1px solid #DDA0DD;"
{{#if: {{{aunts and uncles<includeonly>|</includeonly>}}} | {{!}} '''Aunts and Uncles:''' {{!!}} style="text-align:left;" {{!}} {{{aunts and uncles}}} }}
|-
|- style="font-size:2.5; vertical-align:middle; border-bottom: 1px solid #DDA0DD;"
{{#if: {{{cousins<includeonly>|</includeonly>}}} | {{!}} '''Cousins:''' {{!!}} style="text-align:left;" {{!}} {{{cousins}}} }}
EDIT: Technically you could also just add a field into the template on the page for |relatives, but this only fixes the symptoms and not the root issue