Not for the future, for right now, before you fit the new board and fry it as well.
Resistors generate heat in operation.
If yours is generating so much heat that it smokes then it is doing way too much work.
The extra load could be because something else on the board is blown or because there is a wiring fault in the mower.
The JD manual has an extensive electrical section that tells you exactly how & where to test every part.
You are there with your mower & I am here with my computer.
All I know is you hooked up a set of jumpers & then your board started to smoke.
No where near enough information to offer any useful advice other than what I have already done.
If I had a manual for your mower I would have copied the electrical section & sent it to you however I only have the smaller 172-188 manual as I buy them as required to fix my customers mowers.
as you bought the mower used that adds another lever of complexity as neither of us have any idea what & if the previous owner has done to the wiring.
There might even be a specific method to jump the tractor, have you read the owners manual ?
They can be viewed online for free & might be downloadable