Can horses eat apple cores?

Yup. In fact, they like them. It's perfectly fine to eat an apple and give the horse the core. Although apple seeds are toxic, the amount that would be needed to hurt an animal the size of a horse is huge. Also, the seeds tend to be passed intact.

Some people do say that apple cores can get stuck and cause choke, but I've never seen it. Horses will also eat and enjoy crabapples (which are generally too bitter for humans to handle without adding sugar).