In some cases it doesn't make much sense to create a new class just to do sorting.
Here, is a function that can be used to sort an any number of arbitrarily typed lists (List<?>) based on a key list (List<T implements Comparable>).  Ideone Example here.
Usage
Here is an example of how you can use the function to sort multiple lists of arbitrary types:
List<Integer> ids = Arrays.asList(0, 1, 2, 3);
List<String> colors = Arrays.asList("blue", "yellow", "red", "black");
List<String> clothes = Arrays.asList("shoes", "pants", "boots", "coat");
// Sort By ID
concurrentSort(ids, ids, colors, clothes);
// Sort By Color
concurrentSort(colors, ids, colors, clothes);
// Sort By Clothes
concurrentSort(clothes, ids, colors, clothes);
Output: 
// Sorted By ID:
ID:      [0, 1, 2, 3]
Colors:  [blue, yellow, red, black]
Clothes: [shoes, pants, boots, coat]
// Sorted By Color:
ID:      [3, 0, 2, 1]
Colors:  [black, blue, red, yellow]
Clothes: [coat, shoes, boots, pants]
// Sorted By Clothes:
ID:      [2, 3, 1, 0]
Colors:  [red, black, yellow, blue]
Clothes: [boots, coat, pants, shoes]
Code
An Ideone Example can be found here which includes validation of parameters and a test case.
public static <T extends Comparable<T>> void concurrentSort(
                                        final List<T> key, List<?>... lists){
    // Create a List of indices
    List<Integer> indices = new ArrayList<Integer>();
    for(int i = 0; i < key.size(); i++)
        indices.add(i);
    // Sort the indices list based on the key
    Collections.sort(indices, new Comparator<Integer>(){
        @Override public int compare(Integer i, Integer j) {
            return key.get(i).compareTo(key.get(j));
        }
    });
    // Create a mapping that allows sorting of the List by N swaps.
    // Only swaps can be used since we do not know the type of the lists
    Map<Integer,Integer> swapMap = new HashMap<Integer, Integer>(indices.size());
    List<Integer> swapFrom = new ArrayList<Integer>(indices.size()),
                  swapTo   = new ArrayList<Integer>(indices.size());
    for(int i = 0; i < key.size(); i++){
        int k = indices.get(i);
        while(i != k && swapMap.containsKey(k))
            k = swapMap.get(k);
        swapFrom.add(i);
        swapTo.add(k);
        swapMap.put(i, k);
    }
    // use the swap order to sort each list by swapping elements
    for(List<?> list : lists)
        for(int i = 0; i < list.size(); i++)
            Collections.swap(list, swapFrom.get(i), swapTo.get(i));
}
NOTE: The running time is O(mlog(m) + mN) where m is the length of the list and N is the number of lists. Usually m >> N so the running time is not more significant than sorting only the key O(mlog(m)).