How to filter Search API index view by taxonomy parent
I just figured this out the answer was simpler than I thought. 

I was trying to use contextual filters &amp; relationships like other answers said but what worked was just simply adding the Taxonomy Field to Filter Criteria then selecting the desired parent. That's it. No need to add a relationship to it.

Edit:
There is still a problem: it doesn't work when using a grouped exposed filter it gives this error: 


  A value is required if the label for this item is defined.


Log Message:


  PHP -  Warning: trim() expects parameter 1 to be string, array given in ...


I tried changing the filter to Combined field filter with the Taxonomy as a field so the input would be string but I still get another error.


  PHP - Error: Call to undefined method ...