Hah! Yeah, right. I've done 3 imports - 2 I picked up, and 1 I used a shipping company, and they were one of the worst groups of people I've ever dealt with. Even if I could tolerate talking to them again, I'm sure they wouldn't know the answer.
AFAIK, most vehicle shipping companies have the "running" restriction, because they use drive-on trailers only, therefore the car needs to be driven on. For a parts car, you'd probably be looking more at a non-vehicle shipping company, like Fedex Ground or something. I'm just guessing here, though.
Best bet is to call CBSA or RIV. I haven't got time now, but if I get a chance later I will and post what I learn.