In Tsunami's recent review of the UK's
Cornwall and West Devon Mining Landscape, he makes the following observation: "I don't know how common this practice is in other countries, but off the top of my head I can't think of any National Parks in the US where train runs through, except some tourist trains / steam locomotives."
In that, he's pretty accurate -- most of the US national parks, and certainly all of the ones that are World Heritage Sites, don't have the
Amtrak passenger rail service running through them. One of the closest near misses would be Glacier National Park, whose southern border quite purposefully ends right before the rail line served by the Empire Builder route, which runs from Chicago to Seattle.
That said, most of the national parks do have an early history with the railroads, which served the purpose of getting tourists to remote areas before the United States road network was fully developed. For instance, the former Northern Pacific Railway used to run a route to Yellowstone National Park's
northern border. The
Grand Canyon Railway used to take passengers right up to the park at Grand Canyon Village; it is now run as a tourist train.
If you do hope to travel through national parks via the United States' Amtrak network, your best opportunity would be some of the newest national parks in the system (which will likely not ever be World Heritage Sites). Several of Amtrak's routes run through
Indiana Dunes National Park, near Gary, Indiana, en route to Chicago. Additionally, Amtrak's Cardinal route, from Chicago to Washington, DC via southern West Virginia runs through
New River Gorge National Park.