I certainly did it in the past, and my parrot seemed to enjoy it, but I wouldn't do it today.
First, by going in the box, your bird is basically going into full hormonal drive thinking he or she is making a nest. Like I said, I let my parrot, Sam, do that for years, and he loved it, but I've since read that, unless you are trying to breed, it isn't a good idea to stimulate their hormones in that way. I don't know what to think about that.
But if I were to give my parrot a box to play in/ destroy, I would not give a shipping box. Think of all the things that are in an average UPS truck, including things that are very toxic to parrots. Then think about the fact that they need to fumigate those trucks to deal with bedbugs and other pests. You may not see or smell these chemicals on the box, but paper is highly absorbent. So not seeing it on the outside is no guarantee it isn't in the material. Finally, there is a bunch of nasty chemicals in cardboard itself that you might not want your parrot to be chewing on.
In short: I did it for years, and my parrot enjoyed it; I wouldn't do it now. But I'm a risk averse person in general, and I'm feeling especially risk averse now that I feel that I contributed to my parrot's death.