Tzaar baalei chayim means torturing an animal for no reason but sadism. That’s why it’s forbidden; it promotes a bad middah, and one who enjoys being cruel to animals will end up being cruel to people.
But if it’s done for a reason, even a very slight one, it’s permitted. For instance, although nobody could be so cruel as to pull a feather from a living bird to use as a pen, al pi din it’s permitted, because you’re not doing it to hurt the bird, you just want the pen and the bird’s pain is collateral damage.