Is it still possible to take the train to the Big Game? Does Caltrain still stop near the stadium? The Caltrain website makes no mention of a stop at Stanford, and the Stanford website makes no mention of using the train to get there.

it's an easy 20 min walk from palo alto cal train station.

https://www.caltrain.com/location/stanford-stadium

Sounds like they are not using the stadium station this year due to construction.

You can park at any number of Caltrain stations.
What I do is park at Mt. View so that I'm riding within the same zone and it doesn't cost as much.
Also Mt. View has a lot of parking spaces.
It costs $2.75 to park and $3.20 to ride if you have the clipper card. $6.40 round trip.
Be aware that Caltrain runs only 1 time per hour on weekends and the train comes to Palo Alto @ 7.02 pm and 8.02 pm so, with a 3:30 start, if the game is only 3 hours long, you need to get on your feet right away to catch the earlier one. BTW, all of this is assuming you are coming from south and returning south.

Looks like they decided to run trains all the way to Stanfurd.

https://www.caltrain.com/news/caltrain-will-provide-big-game-service-cal-vs-stanford-saturday