While I have no problems to connect to my localhost database that way:
localdb = MySQLdb.connect( host="127.0.0.1",
onlinedb = MySQLdb.connect( host="212.227.000.000" ...
onlinedb = MySQLdb.connect( host="212.227.000.000/phpmyadmin" ...
OperationalError: (2003, "Can't connect to MySQL server on '212.227.000.000' (10061)")
It sounds like
212.227.000.000/phpmyadmin is the URL of PHPMyAdmin (the thing you open in the browser). If so, the database may not be hosted on the machine with IP 212.227.000.000. You should check how PHPMyAdmin connects to the database. If PHPMyAdmin connects to 127.0.0.1, that probably means the database doesn't listen on the external IP address, and can't be reached over the network.
If you have ssh access to 212.227.000.000 you can check that with the
$ netstat -pant | grep LISTEN | grep 3306 tcp 0 0 0.0.0.0:3306 0.0.0.0:* LISTEN -
The 0.0.0.0 above indicates that MySQL is listening on all IPs, and barring any firewalls, you should be able to connect to the database.
Otherwise, if it says
127.0.0.1:3306, the database can only be accessed from the machine itself and not over the network. In that case you can use an SSH tunnel.