Child Categories Referencing Multiple Parent Categories

This might be a weird question, but I am in need of having multiple child categories to have multiple parent categories, similiar to how posts can belong to multiple categories. Along with this, I need to be able to list ‘only parent’ categories, as well as ‘only child’ categories. I am new to WordPress, so I might be overlooking an obvious option. If it requires me to hack away, I can do that, I just want some input from you all.

Thanks