First of all, your mysql configuration needs to be changed that it doesn't listen only on localhost, so head to my.cnf and comment out bind_address/restart mysql. my.cnf location depends either if you're on centos /etc/my.cnf or debian/ubuntu /etc/mysql/my.cnf.
After that, it should work (in newer plesk versions you can define user connectivity and permissions per user, but since your website is connection to server A, I don't think you can define that-there). The simplest test would be to connect command line - shell and try to login with your user:
# mysql -u your_user_on_weba -p databasename -h IP_or_hostname_of_website A
you execute the above from web C
Good luck