@mainegolfur - there are a couple of thing right off in regards to your issue... the seat switch and the parking brake switch will not stop the engine from turning over (they just cause the fuel solenoid to shut off)... so they are not your problem...
I would check to see if you are getting any voltage at the Starter Solenoid when you turn the key to START... If you see voltage when you turn the key to the START position but it is less than 12V this will definitely cause an issue... If you don't have voltage there when the key is turned to start, report back....
Also NH TC's are known for battery cable issues.... Have you tested your battery cables? If not I would....
This is some info that I have posted previously that may help with the diagnosis....
Cable Test:
Older New Holland battery cables often corrode inside the terminal where it is sometimes impossible to see; not sure it was an issue with your particular yr / model... You will often have voltage so everything seems good. (gauges & all are working). So you will have voltage but they
will not supply the required Amperage to start the machine.
You can test if there is an issue with your existing battery cables pretty simply.
Remove the negative battery cable & then take a negative jumper cable wire and place it on the bare negative post of your tractor battery. Connect the other end to a solid clean ground of the tractor (bare metal location). Try starting the tractor & see if that makes a difference...
If that doesn't change anything you can do the same process on the positive post. Disconnect the Negative jumper from the Battery terminal. Then using your positive jumper cable attach one end to where the positive battery cable connects to the starter (connect this end first), and the other end to the positive battery post (last). Make sure the jumper cable is not able to arc to near by metal. Now reconnect the Negative jumper cable to the battery & try starting the tractor again & see if that changes anything...
Doing this process you basically bypassed the battery cables on the machine, and then can troubleshoot from there.