I don't think the age given as appropriate on the movie package is to be taken as gospel truth. The reasoning behind that is often very dubious, especially when it comes to the ages from when nudity is deemed appropriate and when violence is ...
A film like Schindler's List is likely to be emotionally upsetting, but it is not likely to give your child wrong impressions about how the world is - and that is something I would be far more concerned about.
The difference between 13 and 15 is much smaller than between 3 and 5 or 5 and 7. I'd wager there are teenagers of those ages who are pretty similar in maturity, or even 13 year olds who are more mature than some 15 year olds.
Of course, if you know that your daughter is very sensitive and might be very upset, you'd choose not to have her see it, but if she herself says she wants to, she can likely cope.
Films like this one are ones I only watch very seldomly, because even an adult doesn't enjoy the emotional impact. They are educational, but not entertaining.
The greater danger for children's psyche comes from films that are pornographic, degrade women, glorify violence or, you know, have racist implications while being entertaining enough to make you want to see more.
If a child sees Schindler's List too early she is not likely to search for similar films and watch those and become more and more depressed.