The color of the pony has a double meaning.  The first pony is red because that is a description of the color.  The second one is red when it is born, because the mother had to die.

When Jody first meets the pony, there is quite a contrast between them.  Jody is obedient and immature, and the pony is quite a bit rough around the edges.

Its tense ears were forward and a light of disobedience was in its eyes. Its coat was rough and thick as an airedale's fur and its mane was long and tangled. (Ch 1, p. 9)

Jody is thrilled to get his first pony.  It is a sign of manhood.  He is told it’s his if he can take care of it and break it.  But the pony dies, and Jody cannot save it.  This is a defeat for Jody, and he begins to lose his innocence.

The title also applies to the second pony, which is born covered in blood because they have to smash its mother’s head in to save the pony.  Jody tries to be happy to have another pony, but this pony is tainted by the blood.

He tried to be glad because of the colt, but the bloody face, and the haunted, tired eyes of Billy Buck hung in the air ahead of him. (Ch 3, p. 72)

Both ponies are tainted in some way.  The first is ill, and the second was born but its mother died.  Each pony ushers in another phase of Jody’s life.  He begins to grow up as he realizes that life is not full of the possibilities he thought.