If you own the divine horse "Tarpan" (which is gotten for free as a level 10 trophy reward), then parents have a chance of passing their coats on to their offspring. Although the game has never officially stated how much this chance is, testing shows that it's about a 25% chance of the foal inheriting a coat, or 12.5% per parent.
If you don't own Tapan, then a foal's coat is chosen randomly from among those its breed can come in. Each coat has a certain chance of occurring in each breed. You can view these chances by going into Directories, then into the page of a horse breed. It'll list all coats that can occur in that breed, and the chances of them occurring.
