Add menu links to custom generated menu
The problem is that you're not specifying what menu your links belong to. Instead you are setting the parent, but I think the parent doesn't exist in your cast. This works for me:

custom_module/system.menu.my-custom-menu.yml:

langcode: en
status: true
dependencies: {  }
id: my-custom-menu
label: 'My Custom Menu'
description: ''
locked: true


custom_module/config/install/custom_module.links.menu.yml:

custom_module.custom_link:
  title: 'My Custom Link'
  weight: -10
  route_name: custom_module.custom_route
  menu_name: 'my-custom-menu'