Finalists got preference, but freshers like me could enter a kind of lottery, and if any spaces were left over, we could keep our car in hall. There were also passes needed to drive on campus, but I had the after hours one, the one everyone who got a parking space in hall got.
The idea was that there wouldn't be the chaos of a free for all with more cars than spaces, so they only issued permits for the spaces available.
The prospectus actually said "Freshers are not normally able to keep a car at the halls of residence", so even for the 1970's, I got lucky. The fact that there were fewer cars around probably helped.
It seems to be a free for all now, with residents near universities often complaining about students using their streets as long term parking. Clearly this is because students bring more cars than there are spaces on campus.
Reading university may be unusual in that it has a large campus and many of the halls were on that campus at the time, rather than dotted around the town. This meant that it was easier to police the rules about students bringing their cars if there wasn't the space for them.
Now it seems many students rent houses and student flats in blocks built outside of campus, and not necessarily operated by the university. This will make it much harder to police students in their free time, and restrict whether or not they bring their cars to their lodgings.
It's also much harder for students to afford cars. The car isn't the problem, it's the cost of insurance, and what was often done back in the 70's has been the subject of a crackdown by the insurers, who insist that it's the person who drives the car the most that needs to be the primary insured driver, rather than accepting that the owner/keeper is the primary insured driver, and anyone else is just a named driver.
My car was legally my mum's still (she sold it to me at a deep discount, and bought my brother a motorbike). Once I got to university, I found the specialist insurer Endsleigh, and so I officially bought the car off Mum by transferring it into my name. At the age of 18, my first premium was around £60-£70, and I could then start to build up my own no claims discount.
Now, some 18 year olds are getting cars for a similar price (£250), but are getting quoted several thousand quid for insurance.
........... of course, all this meant that if we had the internet back when I was 18, I would have had ID for the online casinos, and could have played almost anywhere (except Club World of course, being a student at the time

).