How to get and display menu items except the parent item?
You can do this by configuring the Menu levels on your Menu Block and placing that Block in a region of your theme. Additionally configure the Visibility of the block so it only appears on pages where you want it. 

You can learn more about the Menu levels configuration here: Drupal 8 Navigation Menu Blocks configuration. 

In your case you want the Initial visibility level and Number of levels to display to be either: 

1 and 3 

OR 

2 and 3

AND

you'll want the Expand all menu items to be checked, unless you have all the items Expanded under Menu settins.