I have a John Deere LT160 42" cut that the blades would stay engaged when the lever was disengaged. I replaced the brake pads on the spindle pulleys and the blades still stayed engaged. I could turn the engine off and once the engine was restarted the deck would not be engaged until the lever was activated but it still would not disengage unless I turned the engine off. I removed the deck, and sand blasted the breaking surface of the spindle pulleys to add a higher friction surface. I also used steel wool to make sure the belt surface of the pulleys were free of debris and had a smooth surface. I then sprayed Glide Coat into the belt groove to make sure that the belt was not being constrained in the pulleys. The deck belt was brand new JD belt and was the correct belt. In a nutshell I increased the friction of the braking surface and decreased the friction inside the pulley groove as much as possible. Once all the deck was reinstalled on the tractor the deck engages and disengages with the lever as designed.

 

I would guess that there is a return spring that has broken and isn't forcing slack in the belt when you disengage the deck after having it running. Once it is shut down, the lack of drive tension allows the belt to slip on the engine pulley.

I would take a look at torsion spring #2 in the deck parts diagram.

 

The issue is corrected. The JD does not have springs for the brake pad. The brake pad is mounted to spring steel that is bolted to the deck and has a connecting rod that pulls the brake pad off of the spindle pulley when the blade lever is engaged. Unless the deck is engaged the spring steel keeps the pad against the braking surface. The Torsion spring is present and is working correctly.