Custom query loop for terms having a parent child relation

Does the “Parent” or “Child of” fields in QUERY only accepts the ID number?
How to use it for dynamic data?

I am building a loop for the terms to show up in a list type layout on the archive pages.
I want it to load only the related terms of that archive. For example, I have two parents categories, each having dozen child categories under it. I want to show the related child categories on the parent category archive pages.
But i could not make the query loop dynamically do that. I am not able to get the “parent” and “child of” fields to do it for me.
Any help?

This is also discussed here. It should resolve {term_id} in the parent section, but it is not working for many.

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Yes that must be the possible solution but {term-id} is not working for me.