Try setting fire to a horse and see the legal results. Try making a species go extinct, even pests. So, there are some rights - and arguments for even stronger rights.
I hunt and fish, so I'm not a granola munching hippy. Yet, animals still have rights. I can hunt and kill a deer. I can't hunt and kill a deer by intentionally causing excessive pain. There are laws against that. Those laws are the animal's rights.
I didn't say we have obligations (or really duties) _to_ animals but _in respect to_ animals. This is an important distinction.
Animals do not have a right not to suffer, but it is immoral to be cruel toward them. I can, legitimately, cause suffering in an animal if it is an unfortunate effect of, say, acting for an appropriately commensurate human good. I cannot morally do the same to a human being.
What is worth focusing on in this domain is the unprecedented cruelty inflicted on animals in our times where, e.g., cosmetics, fashion and agriculture are concerned.