Even if they agree today, I'd not be surprised if they "magically" find some issue in a few months time that voids some portion of the motorplan, or if they refuse to extend plan (not sure what the real point is of getting a car with just a few months of plan left if you're not going to extend? depending on when in 2020 the car is from).
@Op, this is a shitty situation but I'm relatively confident that if you do take this car, with what we've seen here, it will bite you in the future. We've already discussed how these "independent" checkers are anything but. If you want to truly have a car inspected you're better off asking a knowledgeable forumite to have a look and take as many pics as possible. I haven't followed all of these threads you've had, but TBH if this car is for your folks I really don't see why you'd hunt for a 4yr old BMW (unless it was the deal of the century, which this clearly isn't). Old toppies want a car that runs, isn't too thirsty and can be serviced affordably. This does not tick those boxes.