To access a parent element ID, you can use:
var elementId = elements[i].parentNode.id;
See here for more info: Getting the parent div of element
To find the nearest DIV parent, you should be able to do something like this:
var elements = document.getElementsByClassName(id);
for(var i=0; i<elements.length; i++){
    var parent = elements[i].parentNode;
    while (parent.localName != null && parent.localName.toLowerCase() != "div") {
        parent = parent.parentNode;
    }
    if (parent != null && parent.id != null) {
        var parentId = parent.id;
        alert(parentId);
    }
}
EDIT: Just added some null checking, in case a class is NOT contained in any DIV.
Here is the test I used:
<html>
<body>
    <div id="asdf">
        <a href="#" class="blah">1</a>
        <a href="#" class="blah">2</a>
    </div>
    <div id="asdf2">
        <span>
            <a href="#" class="blah">3</a>
            <a href="#" class="blah">4</a>
        </span>
    </div>
    <!-- These two won't cause an alert since no DIV surrounds them -->
    <a href="#" class="blah">5</a>
    <a href="#" class="blah">6</a>
    <div id="asdf3">
        <span>
            <a href="#" class="blah">7</a>
            <a href="#" class="blah">8</a>
        </span>
    </div>
    <input type="button" onclick="run();return false;" value="Run" />
    <script>
    // This will cause 6 alerts: asdf, asdf, asdf2, asdf2, asdf3, asdf3
    function run() {
        var elements = document.getElementsByClassName("blah");
        for(var i=0; i<elements.length; i++){
            var parent = elements[i].parentNode;
            while (parent.localName != null && parent.localName.toLowerCase() != "div") {
                parent = parent.parentNode;
            }
            if (parent != null && parent.id != null) {
                var parentId = parent.id;
                alert(parentId);
            }
        }
    }
    </script>
</body>
</html>