Not with plain CSS you need some form of script to notify the parent that the child is being hovered(eg.):
<div id="parentId" class="parent">
<div id="childId" onmouseover="doOnMouseOver()" onmouseout="doOnMouseOut()" class="child">
</div>
</div>
<script type="text/javascript">
function doOnMouseOver() {
var parentNode = this.parentNode;
var newParentClass = parentNode.getAttribute('class') + 'child-beeing-hovered';
parentNode.setAttribute('class', parentClass);
}
function doOnMouseOut() {
var parentNode = this.parentNode;
var newParentClass = parentNode.getAttribute('class').replace('child-beeing-hovered', '');
parentNode.setAttribute('class', parentClass);
}
</script>
Note that I've added ids to your html elements so that I can get a hold of them with javascript without making the code unnecessary complex nor using a third party library like jQuery.
Note that you need also to bind onmouseout or otherwise the parent element will keep the new class child-beeing-hovered.
jQuery actually makes your job easier but you should try doing this with javascript at least once.
I hope it helps.