Current leak through solenoid?
 

Had a recent issue on the DD...battery would run down over the course of a week or so if the truck sat parked. Went through all the "normal" checks, even replaced the battery, starter, and neutral safety switch. When the engine did start it cranked a bit longer than it should. It never left me stranded as it would start again after it had run a bit. Finally the starter just clicked instead of start which brought me to the point that it had to be replaced. Now since it has a new starter, the battery stays charged and the engine starts on the first click. I can understand it that the engine starts better but could it be that the solenoid was drawing down the battery while parked? The solenoid is mounted to the starter so there is no separate starter relay. The lead from the battery is connected HOT to the solenoid so there IS a path for current to ground. Has anybody else ever seen this happen before?
As an added note; this has been going on now for a couple of years it's just that lately it has gotten far worse.

Did you try a continuity check from the solenoid hot contact to ground once the old starter was removed?

Sounds possible but first time I've heard of it.

Over time a high current relay or solenoid contact can deteriorate and leave the fine metallic particles bridging the contacts. It will not be enough to fully activate the circuit but can have some current draw.
Wack the solenoid / relay with a hammer. It may knock the bridge off.

I thought about doing that but I wasn't going to dig the starter out of the trash. Did I mention that I was not happy with it...

Kinda what I thought as well...I was thinking that whatever may have got across the contacts may have been something other than metal, perhaps carbon or coil wrapping material. I have never seen anything like this either. The truck has over 200K on the clock and was previous a John Deere service truck so it's hard telling if or how many times the starter had been changed. Old one did not look OEM.