In the state of Victoria, Australia, the expression grey meanie designates a traffic warden, i.e., a person employed to enforce regulations about the parking of motor vehicles.
—Synonyms:
– in the state of South Australia: sticker-licker;
– in the state of New South Wales: brown bomber.

In the expression grey meanie:
– the adjective grey refers to the colour of those officers’ uniform;
– the noun meanie refers to the nastiness displayed by those officers in the accomplishment of their duties.

Kevin Childs mentioned the colour of the uniform in Grey meanies, and how to beat them, published in The Age (Melbourne, Victoria) of Tuesday 3^rd April 1984:

That the men and women attaching the pink punishment are grey makes it all the worse. The traffic wardens of Paris were a stylish sky-blue when I saw them last; they appeared to work for an airline. That is, until they applied the sabot, a metal clog, which is locked on to the wheel of an offending vehicle. To get the key you have to go to the nearest police station and pay your fine.

All the early occurrences of the expression grey meanie that I have found are from the column Briefing, by John Larkin, published in The Age (Melbourne, Victoria).

It was in particular in The Age of Saturday 17^th April 1971 that John Larkin described the Melbourne City Council parking officers’ behaviour:

Who are those people known as the Grey Meanies, the Melbourne City Council officers? Why do so many of them behave as though their work is an opportunity to take out something personal on citizens? What is their hang-up? An ego trip? A power trip? Just plain dislike for other people? Just plain dislike for themselves?
The general experience with them—and this is not just personal, several others have said the same thing—is that many seem determined to make their work as unpleasant as possible.
Who would want to go around punishing other people, anyway?
[…]
You do not want to be paranoid, but the Grey Meanies make it hard.
Sometimes you are only a few moments over on an expired meter time, but this is all you need to be booked.
Sometimes you need stop somewhere only for a couple of minutes, but so often you hear some terrible voice to gettamoveon or weelgetcha.
Why do they have to speak like this? By what right, just because they are in a position of authority? Why can they not give you those couple of minutes’ grace? Are we so regimented by the book? Are we such less people and more machines, to be plugged in and switched off and on?
Worse are the Meanies who tell you not to park at one place and let you park in another, and then go ahead and book you the moment you have walked away from the car.
What gratification is this?
A couple of the officers met over the years have been pleasant people who have given you some chance not to be booked. They have acted as people with a responsibility, rather than arresting policeman, judge, jury and executioner.
But mostly the city scene has become a place of suspicion, sneaking about, a continual searching for chalk marks and people out to get you.

The earliest occurrence of the expression grey meanie that I have found is from John Larkin’s Briefing published in The Age of Tuesday 5^th May 1970:

Here, yesterday, were 151 schoolboys, aged 13 to 17, tramping, trudging, marching 12 miles into town from the Box Hill High School to protest against pollution of the city.
They were doing more than protest—they were doing something about it.
These young men, along the 12 miles they walked, collected all the rubbish they could find.
Up to their elbows in it.
This meant and included everything, on the footpaths, nature strips, road, tramlines, bridges—even in the gutters.
Their collections they placed in large bags and paper Keep Australia Beautiful containers.
[…]
The boys said yesterday the local councils were only too happy to have the rubbish removed.
But, the Melbourne City Council, they were told, were not too pleased. At first they refused Permission. Then they thought about it a bit and said the boys could march—provided they did not carry placards—(some politics may be rubbish, but rubbish being political?)—and provided they did not dump outside Cook’s old place. *
Some of the boys did not take too much to this high-handedness and said they’d dump and be damned.
And dump they did.
They all got to the cottage around 2 o’clock, some staggering under heavy loads.
They piled the stuff around a seat, in a heap. It measured 8 ft. x 8 ft. x 5 ft.
But no Grey Meanies appeared and there were no arrests.
After a bit of bouncing around, the boys prepared to take the rubbish to a nearby dump (legal), where they would separate it into categories. Each lot of cans and chocolate wrappers and the rest of the forest of packaging would be returned to the manufacturer with a note: “We can dispose of the contents, but what are you going to do about disposing of the rubbish?”

* This refers to Captain Cook’s Cottage, located in the Fitzroy Gardens, Melbourne. It was constructed in 1755 in Great Ayton, Yorkshire, England, by the parents of the English explorer James Cook (1728-1779), and was brought to Melbourne in 1934.

In the review of A Royal Commission of Inquiry into Corruption in Victoria, a stage play by John Clarke and Lawyers, Guns & Money, published in The Age (Melbourne, Victoria) of Thursday 6^th April 1989, Kevin Childs mentioned that the authors had named a parking officer Bernadette Kathleen Hitler:

Corruption? In Victoria? John Clarke and his co-writers from ‘Lawyers, Guns and Money’ packed the Police Auditorium to stifling point with an opening-night audience of politicians, legal eagles, public servants and a scattering of VIPs to hear such charges laid.
The noblest of the state’s institutions, such as Melbourne City Council parking inspectors, the police, the Victorian Football League, the Government and even the newspaper you are reading, were held to ridicule, if not contempt, as Victoria joined its sister states in hanging out the dirty washing.
[…]
Just how a strident parking inspector, one Bernadette Kathleen Hitler (Denise Scott), comes to be a royal commission witness and what flows from there is integral to the cunning plot and provides a fine chance for the savaging of the grey meanies.