Birmingham -- the city of Bull Connor and his dogs, of Martin Luther King Jr. locked up in jail and of a church bombing that killed four black girls -- elected a black mayor yesterday. c

Richard Arrington, 44-year-old son of a sharecropper, became the Alabama city's first black mayor with 51 percent of the vote in a runoff with a white attorney-businessman, Frank Parsons.

With all 75 polling places reporting, Arrington had 44,798 votes to 42,814 for Parsons.

The triumph of Arrington, following eight years on the city council, is a high point in the evolution of racial accommodation that has followed the violent civil rights struggles that in the 1960s made Birmingham a symbol of racial strife.

Strangely, it was a racial incident last June that forced Arrington into the mayoral campaign. The incident was the erroneous and fatal shooting of an unarmed 20-year-old black woman by a white city policeman.

The 38-year-old Parsons campaigned on a strong law-and-order stance and a pledge to keep the police department under the control of policemen.

In the closing days of the campaign, Parsons supporters ran newspaper ads portraying Atlanta as crime-ridden as a result of electing a black mayor, Maynard Jackson.

But while blacks apparently went to the polls yesterday in greater proportion than did whites, early indications were that Arrington, who was endorsed by both of the city's daily newspapers, received perhaps 20 percent of his support from white voters.

Such support was crucial for Arrington because whites make up more than half of the city's 129,000 registered voters.

"This is a historic occasion for our city," Arrington said last night, "because the majority of voters have tapped one of the sons of color of our city. The decision says more about our city than all the public relations we can do and all the things we can say."

"The voters rejected a campaign based on fear and rejected a campaign based on sneers."

It was only 10 years ago that the Birmingham city council got its first black member. Asked before the election if Birmingham was ready for a black mayor, Arrington said, "I think it is. The transition it represents creates uneasiness -- I understand that."

It was widely accepted in Birmingham, and acknowledged by Arrington, that a major task facing the city's new mayor would be to heal the racial wounds left by the shooting of Benita Carter last June.

Neither conservative whites nor change-minded blacks were satisfied with city's handling of the shooting. Mayor David Vann, whom Arrington helped elect four years ago by bringing together a coalition of blacks and liberal whites, refused to fire the policeman involved but he did set up a citizens' committee to review the incident.

Arrington also will face many of the same problems mayors do in declining northern cities: deteriorated housing, a downtown that is decaying, and an economy that has largely been passed over by the so-called Sun Belt prosperity of the 1970s.

Arrington has worked on economic development and finance issues on the city council, and generally receives high marks.

But whatever the problems Birmingham faces in the 1980s, there is not the widespread, racially motivated violence of the 1960s. This is largely because of almost two decades of efforts by business and community leaders to help bring blacks into the mainstream of city life -- a change stunningly illustrated by Arrington's election.

First came a change in the city charter, then blacks on the city council, and then the 1975 mayoral race. A coalition of blacks and liberal whites was formed, largely through Arrington's work, and became a domnant force in city politics. It elected Vann, who named a black city attorney, and many saw the coalition as a base for politicians in years to come.

At the same time, the University of Alabama at Birmingham, with its medical schools and research institutes, boomed. The school is, by all accounts, the growth industry that has come to overshadow steel. In 18 years, it has gone from 2,500 to 8,500 employes, and its budget from $10 million to $220 million.

With blacks making up roughly 20 percent of its student body, it has brought them into the classroom with little or no racial problems.

Now that will be Arrington's task in the city at large. "I am a bridge between the races," he says, citing his work on numerous biracial committees. But it will be difficult to please blacks on police conduct while maintaining police support.

Frank Parson, who edged out Vann in an initial mayoral race on Oct. 9 to win a spot in the nonpartisan runoff, said last night, while not yet conceding defeat: "Now we have to unify the city and move forward."