I don't have the correct manual, but I do have one for each of the 25960 and 25191 tractors that are extremely close. I cannot find a safety switch on either schematic that operates with the parking brake, but I suspect mine may have one.
I have been using a pliers to jump the solenoid on this tractor since last winter when the key switch stopped working on me. I checked the two switches I found, and both of them operate perfectly.
Now yesterday, when I went to get an old A/C unit out of my son's garden shed, I decided to let the tractor do all the work and hooked up the yard trailer to it. Jumped on the tractor, hit the switch, and VIOLA! It started with the key!
The only difference I could see was that I had the parking brake set while I hooked up the little trailer. I rarely use the parking brake, because I always try to drive off with it still set.
Are these tractors commonly equipped with a safety switch on the parking brake?