The Iberia Airlines pet policy is as follows:
Pets are accepted in the aircraft cabin and as baggage in the hold.
Acceptable pets in the cabin: dogs, cats, fish, turtles and birds (except birds of prey). Take them with you in the cabin if they don’t weigh more than 8 kg. (including their cage or pet pack).
Iberia does not permit the transport of weasels (ferrets, pine martens, etc.).
You can carry several animals of the same species in the same container if they are small and light enough.
Iberia permits the transport of dogs and cats in the hold with the exception of brachycephalic animals and dangerous breeds.
Taking a pet to a foreign country involves preparation and planning far in advance of your departure date. Requirements for pet travel to a foreign country are set by the destination country. If you are you traveling to another country and bringing along your pet you’ll need to meet the animal health requirements of the country you are visiting.
More information on the Iberia Airlines pet policy