Resolved!

I removed the nested group and applied the conditional test directly to each sub-fields and then updated the field values for each to unique names. Here is the new code:

<?php
/*
Title: Video Row
Setting: sp_theme_settings
Tab Order: 1
Order: 5
*/

/*
 * Description: If the first select field (a conditional), 'show_homepage_video_block,' 
 * is changed to 'yes', then show the following field group, 'show_homepage_video_block.'
 *
 * The field group, 'show_homepage_video_block,' will contain a field (a file) to upload 
 * a static image (a placeholder for the video box; i.e. a video still) as well as another 
 * select field (a conditional), 'define_homepage_video_type,' to allow the user to define what 
 * kind of video asset should be inserted into the video box (either an uploaded video [using 
 * a file field] or an embedded video [using a html field]).
 *
 * However, I can not get the second conditonal to show the correct field, below is
 * my code:
 */

// Conditional Setting: show/hide video block
piklist('field', array(
    'type' => 'select',
    'label' => 'Show Video Block:',
    'description' => 'Show/Hide the video block on homepage.',
    'help' => 'Enable or disable a video block in the footer of the homepage. Preload with a static image placeholder for a better presentation.',
    'field' => 'show_homepage_video_block',
    'choices' => array(
        'yes' => 'Yes',
        'no' => 'No',
    ),
    'value' => 'yes',
));

// Field Group: video values
piklist( 'field', array(
    'type' => 'group',
    'label' => 'Set Video Values:',
    'description' => 'Use this section to upload and manage your video.',
    'help' => 'In this section you can uplaod a video still and upload a video or embed some video code from youtube or vimeo.',
    'conditions' => array(
        array(
            'field' => 'show_homepage_video_block',
            'value' => 'yes',
        ),
    ),
    'fields' => array(
        array(
            'type' => 'file',
            'label' => 'Upload A Video Still',
            'description' => 'This will be the image placeholder.',
            'help' => 'This image should be a screen grab of your video (a.k.a, a video still). Don\'t forget to add a title and a caption to the image after upload (this information will be display in a banner that appears on top of the image).',
            'field' => 'video_overlay_image',
        ),
        array(
            'type' => 'select',
            'label' => 'Video Type',
            'description' => 'Select the video to be added.',
            'help' => 'You can upload a video and host it yourself on this website or embed a video from a third-party source such as youtube or vimeo.',
            'field' => 'define_homepage_video_type',
            'choices' => array(
                'embed' => 'Add Embed Code',
                'upload' => 'Upload A Video',
            ),
            'value' => 'embed',
        ),
        array(
            'type' => 'file',
            'label' => 'Upload A Video',
            'description' => 'Upload your video.',
            'help' => '',
            'field' => 'video_type_value_upload',
            'conditions' => array(
                array(
                    'field' => 'define_homepage_video_type',
                    'value' => 'upload',
                ),
            ),
        ),
        array(
            'type' => 'textarea',
            'label' => 'Embed Video Code',
            'description' => 'Paste in your third-party embed code.',
            'help' => '',
            'field' => 'video_type_value_embed',
            'columns' => 12,
            'rows' => 5,
            'conditions' => array(
                array(
                    'field' => 'define_homepage_video_type',
                    'value' => 'embed',
                ),
            ),
        ),
    ),
));