But is it really exclusively BE? Two of your dictionaries say "chiefly British", which to me means "the normal BE expression, but also used sometimes in the rest of the English-speaking world". I can't remember whether my Canadian cousins say "parking lot" or "car park", for example.

Agree fully with oreg. Also about BE.

I would suggest making sure it's clear that a "car park" only refers to the Parkplatz in the sense of an area in which (multiple) cars are parked, or where (multiple) cars are intended to be parked. This is crucial, because otherwise a Parkplatz is a "parking space". I've seen so many foreigners get this wrong and the result is incredibly confusing.

Parklücke = parking space

A) car park
n. Chiefly British
A parking lot.

The American Heritage® Dictionary of the English Language, Fourth Edition copyright ©2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved.

B car park
n
(Engineering / Automotive Engineering) an area o r building reserved for parking cars Usual US and Canadian term parking lot

Collins English Dictionary – Complete and Unabridged © HarperCollins Publishers 1991, 1994, 1998, 2000, 2003

(usually) AE/CanE: car park = parking lot

Some of the entries have been corrected in the meantime--thank you Doris! In the process, however, some have become inadvertently multi-kulti:
multi-story^AE car park chiefly (Brit.)
As the purely BE variants are there these entries just need to be deleted. The same is true for this one:
car-park routing^AE system -- das Parkleitsystem

One wrong hyphen still needs removing:
car-park -- der Parkplatz --> car park BE

Longman:
car park
noun [countable] British English
1 an area where people can park their cars [= parking lot American English]
2 an enclosed building in a public place where people can park their cars [= parking garage American English]

Macmillan:
car park
noun [countable] British
an area or a building where people can leave their cars for a short time
a parking lot or parking garage

Two more references to prove that the LEO entry
car-park -- der Parkplatz
needs to lose the hyphen.

To make my other point clear: I think it doesn't make sense to list the American spelling of a term that is only used in British English. Therefore, the following entries should be deleted:
* multi-story^AE car park chiefly (Brit.)
* car-park routing^AE system

Just in case it helps, I support all those corrections; there doesn't seem to be any reason not to go ahead and make them.

We're down to just one: except for car-park routing^AE system all offending entries are gone. Nice! :-)