Continuity mistake: Early in the movie when Seabiscuit is shown getting "whipped as many times as possible" to break him, he is being whipped on the right side. Later, Red tells Wolfe prior to the match race not to whip him on the left side because that was where he was whipped as a baby.
Plot hole: Immediately after Seabiscuit injures his leg, the track vet tells owner Charles Howard that the horse will never race again. He offers to euthanize Seabiscuit, even though it was not a life-threatening injury. Even if Seabiscuit was unable to race again, a champion racehorse has more value as a stud horse. It is nearly inconceivable that a random vet would suggest this so early on. If anything, it would be "wait and see."
Answer: I have tried to look this up, and there is no record of Red's parents. Most likely they just died of old age.