In your child theme’s copy of header.php, try replacing:
<?php if ( get_header_image() ) : ?>
<a href="<?php echo esc_url( home_url( '/' ) ); ?>"><img src="<?php header_image(); ?>" class="header-image" width="<?php echo get_custom_header()->width; ?>" height="<?php echo get_custom_header()->height; ?>" alt="" /></a>
<?php endif; ?>
</header><!-- #masthead -->
with:
<?php if ( get_header_image() ) :
if( is_archive() && !is_home() ) :?>
<a href="<?php echo esc_url( home_url( '/' ) ); ?>"><img src="url_of_image" class="header-image" width="image_width" height="image_height" alt="" /></a>
else: ?>
<a href="<?php echo esc_url( home_url( '/' ) ); ?>"><img src="<?php header_image(); ?>" class="header-image" width="<?php echo get_custom_header()->width; ?>" height="<?php echo get_custom_header()->height; ?>" alt="" /></a>
<?php endif; endif;?>
</header><!-- #masthead -->