It depends what you mean by “rights.” If you mean a set of things that only a moral and self-aware creature can have, then non-moral and non-self-aware animals obviously do not have rights. But if by “rights” you simply mean a set of claims against being treated immorally, then lots of animals have rights. Even the Catechism of the Catholic Church claims that it is seriously immoral to cause animals to suffer and die without great need.
But let’s not get distracted by talk of rights. Non-human animals can and do suffer at the hands of cruel human beings. Indeed, each time we purchase meat at a traditional grocery store or restaurant, we contribute to the horrific sufferings and deaths that pigs, chickens, turkeys, and other animals must undergo in modern-day factory farms. We don’t need to claim that non-human animals have rights to question and change our support of these kinds of sinful social structures. Before he became Pope, Cardinal Ratzinger said that these kinds of farms violated God’s plan for creation. It is hard to disagree with him.