I used too move all the time for my GF's job and there's a difference between moving yourself and having a corporation take care of moving. Some moving companies will ask that you have space available for the truck to fit, others depending on what city you are moving to like ( NYC ) will put all of your belongings in a smaller truck the deliver it. Yeah it's a pain in the ass for neighbors when a 70' tractor trailer comes rolling down your street but such is life.
I don't understand how a random neighbor would have the right or even be able to tow a car that is breaking no laws or parking regulations. It sounds like this guy doesn't have any tact. You should call his bluff, while keeping an eye on things to make sure he doesn't knife your tire or something.
I'd park my shit in his fuckiing yard for that bullshit, and by "my shit" I mean a WIDE assortment of garbage and trash. Send em packing!