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I need to override this parent theme function:

    add_action( 'travelify_after_post_content', 'travelify_next_previous_post_link', 10 );
/**
 * Shows the next or previous posts link with respective names.
 */
function travelify_next_previous_post_link() {
    if ( is_single() ) {
        if( is_attachment() ) {
        ?>
            <ul class="default-wp-page clearfix">
                <li class="previous"><?php previous_image_link( false, __( '&laquo; Previous', 'travelify' ) ); ?></li>
                <li class="next"><?php next_image_link( false, __( 'Next &raquo;', 'travelify' ) ); ?></li>
            </ul>
        <?php
        }
        else {
        ?>
            <ul class="default-wp-page clearfix">
                <li class="previous"><?php previous_post_link( '%link', '<span class="meta-nav">' . _x( '&larr;', 'Previous post link', 'travelify' ) . '</span> %title' ); ?></li>
                <li class="next"><?php next_post_link( '%link', '%title <span class="meta-nav">' . _x( '&rarr;', 'Next post link', 'travelify' ) . '</span>' ); ?></li>
            </ul>
        <?php
        }
    }
}

With this in my child theme:

add_action( 'travelify_after_post_content', 'travelify_next_previous_post_link', 10 );
/**
 * Shows the next or previous posts link with respective names.
 */
function travelify_next_previous_post_link() {
    if ( is_single() ) {
        if( is_attachment() ) {
        ?>
            <ul class="default-wp-page clearfix">
                <li class="previous"><?php previous_image_link( false, __( '&laquo; Previous', 'travelify' ) ); ?></li>
                <li class="next"><?php next_image_link( false, __( 'Next &raquo;', 'travelify' ) ); ?></li>
            </ul>
        <?php
        }
        else {
        ?>
            <ul class="default-wp-page clearfix">
                    <li class="previous"><?php previous_post_link_plus( array('order_by' => 'menu_order', 'loop' => true, 'link' => '%title' ) ); ?></li>
                    <li class="next"><?php next_post_link_plus( array('order_by' => 'menu_order', 'loop' => true, 'link' => '%title' ) ); ?></li>
            </ul>
        <?php
        }
    }
}

I can't figure out the proper way to override it in my child theme functions.php file.

I've tried this method:

if ( ! function_exists( 'travelify_next_previous_post_link' ) )
   function travelify_next_previous_post_link() {
     NEW FUNCTION HERE....
   }

That just gives me a white screen.

What would I need to put in my child theme functions.php to replace that particular function?

You can try this

    <?php
    function child_remove_parent_function() {
       remove_action( 'travelify_after_post_content', 'travelify_next_previous_post_link', 10 );
    }
    add_action( 'wp_loaded', 'child_remove_parent_function' );
    ?>

to remove the function to being load.

Or you can increase your child function priority Here is completed solution for you. http://code.tutsplus.com/tutorials/a-guide-to-overriding-parent-theme-functions-in-your-child-theme--cms-22623