I suggest using jQuery. What you are doing right now is easy to achieve without jQuery, but as you will want new features and more functionality, jQuery will save you a lot of time. I would also like to mention that you shouldn't have multiple DOM elements with the same ID in one document. In such case use class attribute.
html:
<table id="dsTable">
<tr>
<td> Relationship Type </td>
<td> Date of Birth </td>
<td> Gender </td>
</tr>
<tr>
<td> Spouse </td>
<td> 1980-22-03 </td>
<td> female </td>
<td> <input type="button" class="addDep" value="Add"/></td>
<td> <input type="button" class="deleteDep" value="Delete"/></td>
</tr>
<tr>
<td> Child </td>
<td> 2008-23-06 </td>
<td> female </td>
<td> <input type="button" class="addDep" value="Add"/></td>
<td> <input type="button" class="deleteDep" value="Delete"/></td>
</tr>
</table>
javascript:
$('body').on('click', 'input.deleteDep', function() {
$(this).parents('tr').remove();
});
Remember that you need to reference jQuery:
<script type="text/javascript" src="http://code.jquery.com/jquery-1.8.3.min.js"></script>
Here a working jsfiddle example: http://jsfiddle.net/p9dey/1/