Solution:
The problem with getting password prompts when not accessing wp-admin persisted even after moving site to root! So the issue was not that it was on a parked domain.
Found solution on updated InMotionHosting page: http://www.inmotionhosting.com/support/website/wordpress/prevent-unauthorized-wp-admin-wp-login-php-attempts#allow-admin-ajax
Solution is to add the following rules to the .htaccess file located in the wp-admin directory:
# Allow plugin access to admin-ajax.php around password protection
<Files admin-ajax.php>
Order allow,deny
Allow from all
Satisfy any
</Files>
Full .htaccess file now looks like this:
ErrorDocument 401 "Denied"
ErrorDocument 403 "Denied"
# Allow plugin access to admin-ajax.php around password protection
<Files admin-ajax.php>
Order allow,deny
Allow from all
Satisfy any
</Files>
AuthName "MySiteUser"
AuthUserFile "/home/my-site/.htpasswds/public_html/wp-admin/passwd"
AuthType Basic
require valid-user