That article is misleading. Cats and dogs have far fewer sweat glands, and they're mostly concentrated at the bottom of their paws, so it's not their primary method of thermal regulation, but they will sweat when they get hot to aid in cooling down.

You will pant to cool down in extreme cases too.

But the article is correct. It's not remotely a primary dissipation method for most mammals. Those animals don't need to sweat to stay alive.