Two across .. even though it is a "family" coaster .. it is still a coaster and you need restraints as the ride does go pretty fast and has some forces from the turns.
Also, your two year old will be tall enough (38") to ride the mine train?
Even if they are .. I would be aware that you probably should help support your child. Even though it is smooth and not a fast coaster .. a little toddler will get bopped around as they just may not be strong enough to withstand the forces.
My son was 3 and was just tall enough to go on .. my wife needed to basically hold him with her arm so he wouldn't move around too much. I doubt he would have gotten hurt, but the movement was pretty sudden for someone so little and not knowing to expect those sudden forces. He still enjoyed the speed of the ride thoroughly though and talks about it to this day.
That being said .. avoid the Barnstormer .. even though height requirement is smaller, we found that coaster to be "rougher" than Seven Dwarves. (he didn't like it even after being on Mine Train). I would avoid it with little children.