It looks that many people want to see the option to copy to categories of parent project into sub project. So far, it is not a default feature of Redmine. If you have so many projects and categories to write them manually, you can write once and use the SQL command below to copy them.
The table issue_categories in Redmine keeps the records of categories of the projects, you just need to copy one project to new project.
INSERT INTO issue_categories( project_id, name, description) SELECT 71, name, description FROM issue_categories WHERE project_id = 61;
Here 61 is the ID of the projects that I copy from and 71 is the new project that I want to copy categories to.