And I've never seen a Mini which didn't have the park mechanism. Which
is not to say it was working on all of them. Even my Morris Minor has
self parking wipers.

Then you'd better get it fixed before your next annual inspection.

From the "Mini comprehensive workshop manual", ISBN 0 909703 16 7,
published by the Leyland Motor Corporation of Australia, 1976,

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Page Nb-1, Section Nb:
THE ELECTRICAL SYSTEM
The information contained in this section refers specifically to new or
modified components fitted to the Mini range coincident with the
introduction of NEGATIVE earth electrical systems and must be used in
conjunction with Section N.

Page Nb-7, Section Nb.8:

WINDSCREEN WIPER
(Lucas type 14W - Permanent Magnet)
Operation
This windscreen wiper, which is produced in either single- or two-speed
form, has two permanent field magnets incorporated in a cylinderical
yoke. The two speed type is fitted with a third brush, to which the
positive feed is transferred when a faster wiper speed is required.

A self-limiting switch is incorporated in the terminal assembly. A cam
on the underside of the gear wheel operates the two-stage switch via a
plunger. When switched off, the motor continues under the control of the
limit switch until the wipers reach the park position, the first stage
contacts open and the motor is switched off. A period of no contact by
the switch follows, then the second stage contacts close to brake the
armature and park the blades in the same position each time.
=====================================================================================

Or perhaps you have the Lucas type 12AUW?

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Page n-1, Section n:
ELECTRICAL AND INSTRUMENTS
The information in this section refers to Australian produced vehicles.
It does not supercede the previous section, but should be used in
conjunction with it.

Page n-3, Section n.3
WINDSCREEN WIPER MOTOR
Lucas model 12AUW, single speed, self switching permananet magnet wiper
motor was introduced on late Mini saloons and vans. The new unit is not
interchangeable with the previously fitted wiper motors.
Description
(snip)
Associated with the terminal assembly positioned below the gearbox are
three fixed contacts, running on an insulated brass slip ring secured to
the underside of the gearwheel.
One of these fulfils the function of providing contact for a
self-limiting limit switch, whilst the others, inconjunction with the
control switch, provide regenerative braking of the armature to give
immediate deceleration and constant blade parking when the motor is
switched off.
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