How can I expose an Amazon RDS database as read only to other partners who access through home broadband?
Presumably your connection is secured so only your office IP can access the database.

With that in mind a secure VPN tunnel that only directs out to the database area would provide security from them accessing any other systems

And from the database perspective his access rights should be secured down to read-only which will prevent him from modifying anything.

See http://dev.mysql.com/doc/mysql-security-excerpt/5.6/en/adding-users.html for more details on creating users, note the GRANT command, GRANT SELECT will allow them select access, so they cannot update, delete, insert or perform any other such actions.

See http://docs.aws.amazon.com/AmazonRDS/latest/UserGuide/UsingWithRDS.IAM.html for more details on RDS authentication if that is required

As for VPN that depends on the infrastructure you have setup, speak to your IT Admins about this