Why do == comparisons with Integer.valueOf(String) give different results for 127 and 128?

I have no idea why these lines of code return different values:

System.out.println(Integer.valueOf("127")==Integer.valueOf("127"));
System.out.println(Integer.valueOf("128")==Integer.valueOf("128"));
System.out.println(Integer.parseInt("128")==Integer.valueOf("128"));

The output is:

true
false
true

Why does the first one return true and the second one return false? Is there something different that I don't know between 127 and 128? (Of course I know that 127 < 128.)

Also, why does the third one return true?

I have read the answer of this question, but I still didn't get how it can return true, and why the code in second line returns false.