Can you replace your 13hp kholer motor on your black deck stx38 mower with a 12.5hp kholer motor off a yellow deck stx38?

I honestly don't see why not. Those Kohler Command air-cooled engines are both very similar vertical crankshaft engines. I wouldn't be at all surprised if the 13Hp is just a slightly larger bore version of the 12.5Hp engine, but I honestly don't know. You'll have to do your own research on that if you really want to know the specifics.

I know that the "yellow deck" STX38 JOHN DEERe Lawn Tractors were the predecessors to the "black deck" STX's. (In other Words, the black deck tractors are the newer ones, and thus, the slight "bump-up" in Hp from 12.5 to 13.0Hp) I don't believe there were a lot of major changes made between the 2 versions, but I don't know for 100%. All I can say is give it a try. They are simplistic enough machines, where you shouldn't have too much trouble modifying whatever it is that you may have to in order to make an engine swap.

*NOTE* The only thing I'd be particularly concerened about is the outer diameter and design (straight or tapered shaft; and key slotted side and/or threaded end) of the output end of the crankshaft, as that will have a direct impact upon using the original electric PTO clutch, if necessary. Of course, if both engines have their original electric PTO clutches, and one ends up being different from the other, you just re-use the one that fits correctly. Also, be sure the pulley diameters are the same or nearly identical, so you don't have any belt tension issues, or gear reduction issues. There is a fair amount of leeway that you have with such a swap, but I'm just putting it out there as something to think about.

With all that said, I think those crankshafts and PTO clutches between the two machines are the same, but I don't know for sure. Again, you can do your own research, via the JD Parts lookups online if you really need to know.