I understand very well that a phrase like "No animals are allowed." is correct English. It means animals are forbidden; they aren't allowed.
But imagine a place or an organization where administration does allow participants or visitors carrying their animals with themselves. I think in modern world it is possible. Then imagine a visitor who asks: "Are animals forbidden here?" Can the director or manager answer: "No, animals are allowed." or "No, animals are allowed here."?
According to my textbook, it's not possible. My textbook says an essentially affirmative answer ("Animals are allowed.") cannot begin with negation 'no' (followed by a comma). But I doubt if this is true.