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I want to create a sub menu, where the parent item is text only (not a link).
I've done so by making the parent item a text separator type menu item and making sub-menu items by selecting it as the parent for other menu items.
Now the URLs of the pages who are sub-menu items of that text separator are /parent-page/page.
How can I achieve my inital task but with the URLs of the sub-menu items just be root.com/page?
There are a number of ways. If your needs are only as you say then then first of these is best.
Use the direct alias plugin https://extensions.joomla.org/extension/direct-alias/
Make the real menu items in a hidden menu and then add menu aliases to the visible menu
install an sef component like artio JoomSEF which lets you override every url
