You can try a sports pitch mix, but really, turf of any description is not able to withstand vehicles driving over it when it's wet, and you'll still end up with the same mess you've got now, so I'm not sure the extra cost of using something harder wearing is worth it. It will withstand foot traffic a little better, but not enough to make it a worthwhile investment, and any ordinary, dry turf will tolerate the occasional incursion by vehicle wheels, but if it's very wet, having sports grade turf will make no difference I'm afraid. This is equally true of any other type of planting.
I live opposite a sports field - currently, it's a mess from lots of rugby played in very wet conditions. What you're maybe not aware of is just how much work it takes by the groundsmen to repair and keep the turf in good condition, it's a full time job, and the same would be true of your roadside verge.