You wouldn't really want players to use platform elevators because the player collision physics are bugged.
If it's just the payload it's a matter of simply matching the speed of the payload with the elevator and firing them to move at the same time. No real need to parent it or the elevator to each other.
Take a look at sdk_pl_goldrush in your sdk_content/tf/maps folder and you can see that it's just a matter of putting one path track vertically rather than horizontally. The carts go down little elevators at the end of the first 2 rounds.
additionally, having players "push" the cart up or down an elevator is bad mechanics, and will cause a lot of balance issues in the game.