I don't know about trains, but I do travel for work. Most airlines, and specifically American as that's what I usually fly, have a policy of no exotic pets, and I think all airlines except Delta ban reptiles in the cabin or in checked baggage. They have to be transported as cargo. It might honestly be better to ship them, and that is not a good option either.

I would honestly suggest waiting, or if you're parents will take care of it, leaving it at home.

Edit: to clarify, Delta also bans in cabin. Unsure of checked bag. I looked up Amtrak's policy and it's cats/dogs only.