Given that it is a convention we can certainly extend this convention to include other species, perhaps for reasons that are not exactly the same as the reasons we extend them to humans. Perhaps these would be more focused on their ability to suffer, etc.
Sure, we could extend it to animals, or rocks, or the ocean. But it doesn't make sense to do so.
Since most people do, in fact, eat meat; it's clear that most people would not support granting animals equal rights with humans.
But not in terms of their right to life or suffering. And practically speaking pet owners treat their pets with a similar morality. They control facets of their pets lives that they would not another adult human's but they do consider their pets as having rights to life and freedom from suffering. Attempts to go against these rights by another person will be met with unbelievable resistance.
Most reasonable people do not treat their pets the same as children. I grant that some do. But those people are generally considered to be a bit "off".
For instance, I'd spend every penny I have to save the life of any of my children. But I wouldn't go much above a couple grand for pet (if that).
To say we shouldn't seems odd too. If we want to where does this meta-ethics come in to say we shouldn't here but it was OK with humans. If the issue is practicality, well, again, one can point to numerous examples animals being granted rights working out just peachy for really a rather large number of people.
Rights are a way for members of an intelligent social species to deal with each other. In the absense of a system of mutually agreed upon rights, we exist in a state of constant warfare. With people constantly fighting, not much progress can be made. It's extremely inefficient. So we agree to treat each other fairly, to respect each other's "rights".
There is no war between man and cows or pigs. Quite the contrary. So there is no reason to extend the concept of "rights" to include them. Man would lose an important source of food; and the cows and pigs would soon become extinct once they were rendered of no value to humans. It would be a lose-lose proposition.