Parrot mouth is a term used for an overshot jaw in horses. The overshoot, when severe, often resembles a parrot's beak when the lips are pulled back.
In general, it's not visible in newborns, but becomes apparent between 1 and 6 months of age. It's not directly heritable, and is as often caused by trauma or illness stunting growth of the lower jaw as by genetics. When caused by genetics, it's a result of breeding horses with very different skull shapes together, rather than a faulty gene.
Parrot mouth can be very minor or it can result in the incisors not meeting at all. This causes the incisors not to be worn down properly and then grow into the opposite jaw (as can sometimes happen with rabbits). They can also restrict the normal forward and background motion of the jaw, which causes problems eating and interfers with the action of the bit. It's also common for there to be problems further back in the mouth.
Parrot mouth is fixed by specialist dental care while the horse is still growing. This can, in severe cases, include braces. (Yup. Braces for a horse - but they're very expensive and not all breeders can afford them). If not treated it can shorten the animal's life by increasing tooth wear.