Exposed Filter of Child Terms on Parent Page
You can create a separate Views of type Taxonomy term (not content!) display Block that is showing taxonomy term names, add the Relationship to the parent term and require it, and a Contextual filter to taxonomy term ID using that relationship. 

Test it by entering parent term ID into Preview, you should see the parent's children. 

Edit the Format to set the fields to display inline, or use CSS (later) to make them appear horizontally. 

Then in your Content (original) Taxonomy Term views add View area into Header and choose the above Block with Inherit contextual filters option ticked.