No, the oil pump drive is different. You can buy K13 oil drive parts. Also there are some parts machined differently in the K13 clutch basket and an another part that promotes increased oil flow to the plates which is important. A point of nastyness in the K12 clutch that the new model helps change. There are Barnett plates that can be used in both clutches if you don't want to pay the (900$ for the OEM laminar pack). The only thing you need to do is remove the two parts next to the inner pressure surface that fit into the larger ID of the first pressure plate, one is a spacer and the other a belleville spring washer intended to help separate the plates in neutral and while shifting.
Mount Vernon, WA.
tire guy, I'll mount your tires.