Coriander is a herb, commonly called Cilantro in the USA. As a very rare allergy it is not well studied and there has been no confirmation of which allergens are causing the reactions.
Allergic reactions are most likely to be caused by Bet v 1-like proteins, which cause reactions in people sensitised to birch tree pollen and give oral allergy type symptoms.
Another suggestion is that plants in the Apiaceae family contain profilin proteins which can cause allergic reactions in uncommon circumstances.
Allergy to coriander is loosely linked to Pollen Food Allergy Syndrome. You may have Pollen-Food Allergy Syndrome if you suffer from dill allergy with oral allergy symptoms to 3 or more of the foods mentioned in the cross reactivity section.
Coriander is in the family Apiaceae, other spices in this family are anise, caraway seed, celery, chervil, cumin, dill, fennel and parsley.
If sensitised to Birch pollen you may have Pollen Food Allergy Syndrome and may also react to Apple, Carrot, Kiwi, Pear, Peach, Plum, Nectarine, Apricots, Cherries, Tomato, Celery, Potato, Parsnip, Pepper, Cumin, Peas, Dill, Fennel, Hazelnut, Walnut, Almonds, Peanuts, Lentils and Beans.