Animals have no rights, except for those that we might establish conventionally by law. We have indirect obligations insofar as cruelty to animals may harm other humans, who own or care for animals, harm ourselves by adopting bad habits, or break our obligations to God. Thomas Aquinas held such a position because he argued that animals have no immortal souls. In a similar vein, Kant argued that animals do not have reason or free will. Human beings are morally equal to animals.
Animals have rights, as do humans.
We have positive obligations to animals. Animals deserve our moral consideration because, like us, they have a capacity to suffer. We have positive moral duties to liberate animals and people from suffering. To do anything else would be unjust speciesism. These sorts of theories argue that we have positive obligations to animals, and they have rights.