EDIT:
As pointed out in the comments, this is a much simpler version:
Arrays.stream(stringArray).mapToInt(Integer::parseInt).toArray()
This way we can skip the whole conversion to and from a list.
I found another one line solution, but it's still pretty slow (takes about 100 times longer than a for cycle - tested on an array of 6000 0's)
String[] stringArray = ...
int[] out= Arrays.asList(stringArray).stream().map(Integer::parseInt).mapToInt(i->i).toArray();
What this does: