Your parents are mistaken, cats and dogs do get along well together. It will be perfectly safe to drop Fluffy over the fence so she can make friends with the Wilson's Bull Terriers.
And, for that matter, Ah, you say that 3+5 is 8, but what stops it being 18? Consider rejecting the arbitrary axioms of mathematics, and you will see that it could just as easily be 235 or 835 or even five million! Hang on, I'll draw you a diagram.
Have you ever realised that people don't really exist? Not even you! Everyone is a sack of little processes, and that sense of being "you" is just one of those processes that thinks it's in charge, like the figurehead on a ship. And some people don't even have that -- they're just zombies! Maybe your parents are just zombies!