I will be renting an ECV for my Disney trip next week due to knee surgery. I really only need it for the loooong walks- I can handle standing in lines or short walks (i.e. in a shop).
What do I with it when not needed? Does it get parked like a stroller? Should i worry about theft?
Yes, you can leave it in stroller parking (or outside a shop) when you want to leave the ECV. I'd suggest bringing something to put on it so you can recognize it if there are other rental ECVs around (a piece of fabric to tie on the handle, would work). It's unlikely it would get stolen intentionally, but someone might grab it thinking it's theirs.
There are some lines you may want to bring the ECV through the line. For example, Soarin' and the Safari both have lines that are 1/4 mile (or longer) one direction with no way to shorten the distance walked (and you'll have to go back the same distance to get back out).
As others have stated, you can park it with stroller parking and take the key with you.
Just beware that your scooter may get moved while you're gone. CMs will periodically tidy up an area and move strollers & scooters around a bit. This threw my Mom for a loop the first time it happened to her and her scooter wasn't where she left it. So don't panic if you don't see yours when you return. Just look around the general parking area and you should spot it. This is where an easily identifiable object on your scooter will come in handy.