How can you show taxonomy child terms on the parent term page?
One way of doing this is with the use of https://www.drupal.org/project/eva which lets you attach any entity to any other entity 

Just for demonstration I have setup 

parent 1
  child 1.1
  child 1.2
parent 2
  child 2.1
  child 2.2


install and enable eva 
create a taxonomy terms view called children with no block or page display



Then add an EVA views display from the dropdown 
You will need to set a contextual filter parent term with following settings

Hide when value not available 
specify validation criteria 
   taxonomy term id: 
      [NAME-OF-VOCABULARY]
   hide view 




and on entity content settings set entity type: taxonomy term and bundle: [vocabulary name]

 

I had to clear cache so it does not hurt to do it and visit any of the taxonomy parent pages you should see the children terms



you can change where the children are displayed at your taxonomy term display settings page