Views (Search API): Search taxonomy parent and display all children
There's a wonderful little checkbox in the Search API workflow tab which enables it.

Head to:

admin/config/search/search_api/index/*your solr server*/workflow

(You'll need to have created a server to get to this.)

And then tick the box "Index Hierarchy".

In callback options select the taxonomy field where you would like searching for the parent to include the child.