Of course they do. Let’s consider what a “right” is. Rights are bestowed by one party onto another party. Animals have rights because we gave them rights; we just need to give them more rights.
There was a time when women had no rights. The Women’s Liberation movement changed that. For example, in the USA women gained the right to vote in 1920. Similarly the Civil Rights movement gave rights to people of color and children’s rights gave rights to children, Prior to the bestowal of rights upon all of the people, women, people of color, and children had no rights. Now, the animal rights movement is recognizing that animals deserve rights too. These may not be the exact same rights as humans have, but they are still rights. Consider, for example, the fact that children have rights, but they do not have the same rights as adults. For example, children do not have the right to vote.
Currently, there are laws against animal abuse. These laws give animals the right to be free from cruelty and impose penalties on humans who violate these rights. The laws against animal cruelty are not merely laws against harming a person’s pet, which may be considered property. These laws also protect stray animals, which are no one’s property.