But the parent also claimed he can make the same money without the degree. Both things can not be true at the same time ("you need the useless paper", "you can make the same money without it").

Apart from that, while the usefulness of some degrees can be questioned, there is a general concept called "signalling" which has actual value. That is you demonstrate certain qualities in a way that are difficult to fake (usually by doing something costly, like getting a degree, or carrying around a huge tail if you are a parrot).