Unfortunately it's not possible to set the default_timezone in the RDS DB ParameterGroups so your attempt was the right direction already.
$ rds-describe-db-parameters default | grep "time_zone"
DBPARAMETER default_time_zone engine-default string static false
To set the global value via SET GLOBAL you need to have the SUPER privilege which is not granted to you as a RDS user.
The only way to set the time_zone is on a per-connection basis
mysql> SET time_zone = timezone;
On my machines I've tried US/Eastern successfully but I got a quite old generation running.
To determine the timezones you have available log into your box
mysql -h yourboxhost.rds.amazonaws.com -u <youruser> -p
and type
mysql> SELECT * FROM mysql.time_zone_name;
You should get a list of installed and valid timezone names you can set on your instance
+----------------------------------------+--------------+
| Name | Time_zone_id |
+----------------------------------------+--------------+
| Africa/Abidjan | 1 |
| Africa/Accra | 2 |
| Africa/Addis_Ababa | 3 |
| Africa/Algiers | 4 |
| Africa/Asmara | 5 |
| Africa/Asmera | 6 |
| Africa/Bamako | 7 |
| Africa/Bangui | 8 |
| Africa/Banjul | 9 |
| Africa/Bissau | 10 |
| Africa/Blantyre | 11 |
| Africa/Brazzaville | 12 |
| Africa/Bujumbura | 13 |
| Africa/Cairo | 14 |
etc...
You have to set the time_zone each time you connect to your database server
For example if you use the php Mysqli extension you can do this
$mysqli = mysqli_init();
mysqli_options($mysqli,MYSQLI_INIT_COMMAND,"SET time_zone = 'Africa/Brazzaville'" );
mysqli_real_connect($mysqli,$host, $user, $pass,$dbName) or die ('Unable to connect');
Otherwise just manually ( in terms of let your database connector do it ) execute the SET time_zone = '<YOUR_DESIRED_TIMEZONE>'
Query right after you've connected to your database
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