Couldn't you do it as a continuation of the parent task. So execute
FameMappingEntry.SaveFameDBMap(toSaveIdentifiers);
as a continuation of a successful completion of the parent task. And then you can wait on the continutation.
var childTask = parentTask.ContinueWith((pt) =>
{
FameMappingEntry.SaveFameDBMap(toSaveIdentifiers);
}, TaskContinuationOptions.OnlyOnRanToCompletion);
And then you can decide if you want to wait on the child task or use another continuation.