Find an array inside another larger array

The requirement of "using just core Java API's" could also mean that they wanted to see whether you would reinvent the wheel. So in addition to your own implementation, you could give the one-line solution, just to be safe:

public static int findArray(Integer[] array, Integer[] subArray)
{
    return Collections.indexOfSubList(Arrays.asList(array), Arrays.asList(subArray));
}

It may or may not be a good idea to point out that the example given contains invalid array literals.

Clean and improved code 

public static int findArrayIndex(int[] subArray, int[] parentArray) {
    if(subArray.length==0){
        return -1;
    }
    int sL = subArray.length;
    int l = parentArray.length - subArray.length;
    int k = 0;
    for (int i = 0; i < l; i++) {
        if (parentArray[i] == subArray[k]) {
            for (int j = 0; j < subArray.length; j++) {
                if (parentArray[i + j] == subArray[j]) {
                    sL--;
                    if (sL == 0) {
                        return i;
                    }

                }

            }
        }

    }
    return -1;
}