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>