You might also say that humans are the parasites of the world, but this isn't the point.
A parasite is an organism that increases its own fitness (reproductive success) while causing a cost to the fitness of another individual. Since all pets obtain food (and veterinary treatment) due to time and money spent by humans, they should be classified as parasites. It doesn't matter that the human perceives a benefit. If a mosquito injected you with ecstasy while it sucked your blood you wouldn't claim that it shouldn't be classed as a parasite.