Alright I'm not mechanic, but do understand cars..
I've googled and looked at a lot of diagrams...

But my mom was told that the rod between the rear tires is rusting and needs replaced..

My dad called it a tube...

2005 Town and Country, (Front Wheel Drive)

I keep finding axles and front ends..

I just don't know what its called to do a decent google search...

 

I think you keep finding axles because, from your description, that sounds like the item you need.

I've seen a fewl cases of the axle rusting through on other forums.

See these threads:
Axle sitting on one frame bushing. Pics included
Removed my rear axle. How do you install a new one? Alignment? Pics included

The axle is Item 10 in this link: 2005 Chrysler Town & Country Bar,Rear Sway and Traction Bar

And Item 17 in this link: 2005 Chrysler Town & Country Suspension, Rear

Your best bet might be a salvage yard.

 

Rod could be rear anti roll bar about £15 UK. Easy & cheap fix. You really need 'real-world', eyeball + photograph what it is to avoid blundering into the expensive world of parts- darts. The metal rod drop links I speak of are irrelevant in engineering terms, its only the knuckle joints on both ends of the rod that wear, cause 'knocking' and need fairly frequent replacement. The rod itself is merely a fixed measure of length - if they are rusty emery cloth till bright metal and grease - or as stated they are as cheap as chips and an easy DIY replacement job.

 

I also think its the rear sway bar he is talking about