No pets do not have rights. You'd be a fool to give them any. you can punish the people who hurt them, but to give them rights they would have to be able to converse with us or we would have to be able to read their minds.
As it is, a human would have to represent them so they are being represented by someone who can only assume how they feel, and we would be subject to laws and punishment thought of by people who all have their own interpretation of what cruelty is.
There are really radical people out there who think we shouldn't fish or hunt animals, eat meat, walk them on a leash, ride, make them work, make them live outside, etc. Maybe you won't be able to swim in the ocean for fear of invading some fish's privacy.
Once you start that ball rolling it's only going to gather speed and mass. So leave the constituional rights to people, making laws to punish people who are cruel is a more definitive and enforcable way to keep things in line.