Chicago, the legendary "city that works," has a small problem: 35 million unpaid parking tickets.

That's a treasure worth more than $500 million, if it can be collected.

Every day of the year, city police write up another 10,958 new parking violations: an average of 4 million summonses a year, for everything from letting one of the city's 31,000 parking meters expire ($10) to parking in a crosswalk ($15) or at a bus stop ($25).

The wads of tickets have choked the city administration, allowing wayward motorists to ignore their summonses with increasing boldness and to get off without paying. Although up to 20 Cook County Circuit Court judges work in Traffic Court, hearing a total of about 1,800 cases of parking woe a day, hundreds of cases are summarily dismissed as the judges try to keep their heads above water.

Friday morning alone, 222 cases were scheduled for the 9 a.m. opening session at the LaSalle Street courts in The Loop, with three more case calls throughout the day. After queuing up at Counter 103, where courtroom assignments were made by harried clerks, the hordes of defendants shuffled off to the maze of courtrooms.

For many, such an encounter ends with unexpected swiftness when defendants whose last names start with the letters A to J are called to the bench, given a stern lecture about abiding by the law and sent home without paying a penny.

"I tell the judges to hear as many as they can," said Judge Daniel J. White, who supervises the court. But it doesn't seem to make much difference.

The backlog of 35 million unpaid tickets dates from 1978 and attempts to reduce it have barely dented the surface.

Two years ago, the city released the names of some 52,000 scofflaws who had ignored up to 150 tickets apiece. Most of them ignored the ignominy and went on parking where they pleased.

"It's no fun to drive in Chicago anymore," lamented Judge White. "The parking situation is impossible."

"The sheer volume of parking tickets creates serious difficulties for Traffic Court," a special Traffic Court task force appointed by county court leaders reported this week. It numbered these shortcomings: "a poor public image of the court system as a whole; overburden [ed] judges . . . who are less able to deal with serious traffic offenses; and . . . possible corruption where individuals or businesses have accumulated large numbers of unpaid parking tickets."

Allegations of bribes and ticket-fixing in the scruffy halls of Traffic Court are at the center of the sensational Greylord undercover investigation of local judicial corruption. Two judges have been convicted, along with 11 other "lawyers, policemen and bagmen," as an assistant federal prosecutor put it. Ten more await trial. Two have been acquitted. New indictments seem likely soon.

The task force, primly noting the steamy atmosphere around the court, observed:

"Hustling of clients by attorneys at Traffic Court appears to be a serious problem. The task force is making inquiries about attorneys who allegedy tell clients falsely they are able to influence a particular judge or pressure clients to plead guilty so the attorney can immediately obtain his fee with as little work as possible."

In desperation, Chicago is contemplating employment of a private firm to collect the parking fines. In this, the Windy City is following the lead of other places, including Washington, Boston, New York and Philadelphia.

The major force in this little-known but burgeoning private-sector industry is New York-based Datacom Systems Corp. It seems to work miracles wherever it goes. Steven Lipsitz, Datacom executive vice president, cites Boston as typical of what can happen when the computer whizzes plug their circuits into the local mountain of scofflaws.

"Boston was collecting $4 million in parking ticket revenues a year and spending $3 million to do it," Lipsitz said. "Our first year of operation there, they netted $22 million -- that was the first year. Now, they are netting 25 to 26 million yearly."

A similar story is told in Washington, D.C. In the late 1970s, pre-Datacom, the city garnered less than 50 cents on every dollar of parking fine. Today, with Datacom as a consultant, the District is getting a 70 percent return, good for about $22 million in fiscal 1984 revenue.

From Chicago, whose coffers have never recovered from the recession, such returns look appetizing.

Chicago's problems were compounded when a ruling in an out-of-state court case forced the city to stop sending letters threatening scofflaws with arrest if they didn't pay their tickets.

"Our presumption is that collections dropped dramatically," said a local parking-ticket authority.

The prescription for change includes widespread towing and use of the Denver boot to immobilize scofflaws' cars, the authority said, adding:

"Then drivers will know the city knows who they are -- and that it intends to collect from them."

Special correspondent Laurie Kalmanson contributed to this report.