Maybe this will help you move forward.
To set the Parent Category via API in Business Central, you're right — it’s not exposed in the default API, so you’ll need to:
Steps:
1. Extend the standard API page APIV2ItemCategories (Page ID: 5450) in your extension.
2. Add the Parent Category field using addafter() or addlast() in your page extension.
3. Make sure the Parent Category Code field is exposed as a GUID (SystemId) if you're working with the OData V4 endpoint.
4. Publish your extension, and test using POST or PATCH to update the category.
If your changes aren’t working:
Double-check that your API page is marked with EntityName and EntitySetName.
Make sure your modified page is included in the published app and is enabled in the Web Services list.
You can also review Microsoft’s base extension here for reference:
https://github.com/microsoft/ALAppExtensions/blob/main/Apps/W1/APIV2/app/src/pages/APIV2ItemCategories.Page.al
Mark below checkbox to make this answer Verified if it helps you.