How was parking in major cities regulated in the early 20th century?

I can add a few points here for context but I'm afraid I can't cite any specific parking laws.

Firstly, I think it's best to remember that a great deal of those who worked and did business in the CBD at the turn-of-the-century commuted to work, either by foot or by rail. This made traffic congestion terrible during business hours but most CBDs quieted down significantly in the off-business hours, therefore alleviating the need for spacious overnight parking as we understand it today.

Secondly, the street did not belong to the automobile in the early 20th century. In fact, up until around 1920, the street was viewed as an operative extension of the public market and a place to promenade and do business. It was not viewed simply as a traffic artery. Directional traffic - i.e. keeping traffic to the right side of the road - was adhered to only arbitrarily and by custom. People felt perfectly comfortable crossing from sidewalk to sidewalk where they pleased. Hucksters, fashionable men and women, newspaper "bo's and 'gals", teamsters, horses, hacks, carts, wagons, and streetcars shared the space of the street. Deliverymen and porters parked their vehicles where they could delivery their load easiest, either on the street or in service alley's where they existed, and then continued on their route. I think it's best to understand that while streets were incredible congested in the early 20th century, they were rarely stagnant. What traffic regulation did exist was easily forgotten or difficult to enforce. Police officers were stationed at major intersections to guide traffic and speed limits were passed to keep traffic to a walking pace. McShane and Tarr comment on the peculiarity of turn-of-the-century street life in their book, The Horse in the City: "There were other rules, but they seemed mostly to affirm custom or common sense - keeping to the right or not leaving horses unattended to the curb. Turn-of-the-century film clips show no example of either custom being violated. Indeed, those clips are nearly devoid of any regulatory presence - there are no signs or signals, and only one traffic police officer is shown" (p55).
Check out this incredible video of Market Street in San Francisco from 1906 to give you an idea of the lively and haphazard street culture of early urban America.

Where parking was most concerned was with the housing of horses. The largest streetcar companies owned incredibly spacious stables, dotted across the city at the major terminus points of streetcar lines. One of the largest stables was constructed in 1888 in New York City. It was four floors high and housed over 2,000 animals. Indeed, some of the larger horse stables took up entire city blocks.

The irony of the automobile is that it was initially heralded as a savior to the haphazard culture of early urban street life and congestion. And by 1920, as more and more Americans adopted the car as their "go-to" form of transport, cities focused their attention on using public funds to pave roads with asphalt and/or cement, a material that did not agree with the joints and hooves of horses (Cobblestone and dirt are easier on the animals' joints and allow a better grip than smooth surfaces). Furthermore, the streetcar companies found it increasingly difficult to compete with the automobile and the ridership of streetcars peaked in the 1920s, only to decline heavily over the next few decades.

But as many of us know now, the automobile did not eradicate traffic congestion. What the automobile did do, however, was officially turn the city street into a well-regulated, pedestrian-free traffic artery and create a demand for parking that was perviously unknown in the turn-of-the-century city. It should come to no surprise then as the horse declined and automobile ridership rose, a good deal of streetcar horse stables were converted into automobile parking garages in the early 20th century. As for parking meters, those too are consequence of the automobile, with the first official coin-operated meter appearing in 1935 in Oklahoma City.

"The Horse and the City" Clay McShane and Joel A. Tarr, 2007.