Select child when multiple parents with same class - Javascript

I would like to know you if it is possible with pure Javascript to select the child of every parent with the same class. For example:

<div class="parent">
    <a href="#" rel="category tag">Category1</a>
</div>
<div class="parent">
    <a href="#" rel="category tag">Category2</a>
</div>
<div class="parent">
    <a href="#" rel="category tag">Category1</a>
</div>
<div class="parent">
    <a href="#" rel="category tag">Category2</a>
</div>

EDIT:

In this case, I need to select the element of all the and change the background color based on the category: if the category is "Category1" background should be blue, if the category is "Category2" should be red and so on...

Thank you very much.