Possible parent values for parent of links
You can go to the menus structure and click on edit of the menu link under which you want to place the new link. The example from your question would open in this url:

/admin/structure/menu/link/system.admin_config_system/edit


Then you'll find the parent name in the slug before /edit.

But the easiest way is to register the menu link without parent, which would place the new link in the top level of the menu and then drag and drop the link where you want it to be.

You probably need to specify the menu name in the yaml file:

menu_name: admin




In the exported configuration the links you've moved in UI appear in this configuration file:

core.menu.static_menu_link_overrides.yml


where you'll find the parent value where you dragged the link to. You could then use this information for module.links.menu.yml, if you want to deliver your module with this information.