I don't have anything as far back as 10.6 to test with. But if you do, the steps to test are the same for ARD and for the built-in screen sharing client. With the necessary TCP port 5900 port mapping set up (and possibly also eventually with dynamic DNS to allow you to easily acquire the target IP address), you can test the connection with the built-in Screen Sharing client and the built-in screen sharing server.
I've posted some previous write-ups on remote access (see links below), and these might provide you with some background. Alternatives include remote support with Back To My Mac or with one of the screen sharing tools. (You probably won't need to buy the Apple Remote Desktop (ARD) package here, too. The built-in screen sharing client will likely meet your needs. Both the screen sharing client and ARD use the same built-in screen sharing server.) Or as another alternative, one of the commercial screen-sharing services. The do cost a little, but can usually punch through the intervening firewalls and the rest for you, and various of them can enable fully remote access.
Re: Question about Using Apple Remote Desktop over the Internet
Re: apple remote desktop for parent's computer on another subnet?
Re: Problem connecting from my home to work to different stations!
Re: Instructions for managing macs w ARD over internet (no static IP)
Re: Can I Access and Manage my Mac mini at Home from my MacBook Pro While I'm at Work?
There are additional discussion pointers in some of those topics...
I'd be leery of configuring screen sharing or other port forwarding and opening up unlimited remote screen sharing access. Open network ports are now probed by attackers frequently and probably continuously, looking for configuration errors and weak passwords. Screen sharing access is very commonly targeted, too.
Irrespective of all that, upgrade them off of OS X 10.6. That's ancient, insecure and unsupported, it's missing a number of security features, and it won't allow users to perform secure web connections to remote services that require security (banking, etc), and that system is going to be a source of more problems than the remote screen sharing connection. There are problems with software that far back.
When upgrading, I'd get them as far forward as that particular model of Mac supports. The MacTracker tool is handy for determining that; what the most recent version any particular Mac supports.
If you've got the budget for it and if their ISP allows it, I'd look at getting a mid-grade gateway-firewall box and preferably with an embedded VPN server and built-in support for DDNS dynamic DNS. The VPN for the secure remote connection, and dynamic DNS to allow you to find the target IP address as few folks have public static IP. That gateway-firewall box will allow you to do rather more remote management than would otherwise be feasible, too.