Assuming parent really is the direct parent of the boxes, you'd have to iterate and remove each one
function remove_textbox(){
    var parent   = document.getElementById("div1");
    var children = document.getElementsByName("textbox1");
    for( var i=children.length; i--; ) {
        parent.removeChild(children[i]);
    }
}
function remove_textbox(){
    var parent   = document.getElementById("div1");
    var children = document.getElementsByName("textbox1");
    for( var i=children.length; i--; ) {
        parent.removeChild(children[i]);
    }
}
document.getElementById('test').addEventListener('click', remove_textbox)
<div id="div1">
    <input type="text" name="textbox1">
    <input type="text" name="textbox1">
    <input type="text" name="textbox1">
    <br /><br />
    <button id="test">Remove inputs</button>
</div>