Your selector is working correctly, as the active pseudo class, is up when you are clicking on that element, you can check it here, the child_2 element will turn red when you click it's container:
.parent:active > .child_2{
  color: red;
}
  <div class="parent active">
    <div class="child_1">1</div class="child_1">
    <div class="child_2">2</div class="child_2">
  </div>
 
 
Now, what I think you are looking for, is the to select the child, when  the parent has the class .active. And that is a completely different selector.
That is something like this: .parent.active > .child_2, as you can see the following example:
.parent.active > .child_2{
  color: red;
}
  <div class="parent active">
    <div class="child_1">1</div class="child_1">
    <div class="child_2">2</div class="child_2">
  </div>