There are several factors at work. In addition to the things others have said:
Genes which are more advantageous live to reproduce themselves. Since the parent can't acquire fundamental new traits during its lifetime, eventually, it's off-spring and grand-offspring (etc.) will out-compete it with their newer, more advantageous genes. So even if the parent didn't ever die of old age, eventually it would starve, or fall behind the rest of the herd and be the one the predator picks off, or whatever. Superior new generations continue while the primitive progenitor dies.
Once an animal reproduces, it's nearly irrelevant to evolution because any new advantages it did manage to acquire after that would die with it and not be passed along. Therefore, there's little long-term advantage to the parent staying alive once the offspring are mature, so there's little evolutionary pressure for the parent to outlive its usefulness. And evolution seldom wastes resources developing something which has no utility.