Use a temp table in a Table-valued Functions

No.

You can use @table_variables though. Although from a quick glance maybe a recursive CTE might work for you rather than using these child/parent tables at all.

You can use a table variable instead.

DECLARE @Child TABLE (Id INT, Name VARCHAR(255), ParentId INT, ParentName VARCHAR(255))
DECLARE @Parent TABLE (Id INT, Name VARCHAR(255), ParentId INT, ParentName VARCHAR(255))