Parents have been storming the District Building. Local politicians have been warned that their jobs may be in jeopardy. Coalitions are forming across geographic and racial lines.

Their mission: save the public schools, which, finally, are showing signs of academic improvement; restore full funding for public education.

But behind this dramatic and encouraging civic involvement is a deadly serious intragovernmental struggle involving turf, statutory authority and intelligent fiscal planning.

In a nutshell, the issue is this:

The mayor and city council are responsible for setting the limit on the amount of money allotted to the school board. But the board itself has the responsibility for determining how the money is to be spent.

Under normal circumstances, the two provisions are not in obvious conflict. But when, as now, the financially strapped city has to make drastic cuts in its overall spending, a clash is all but inevitable.

From Mayor Barry's point of view, it is impossible to make intelligent cuts in the school system's appropriation without knowing in some detail what effect the cuts will have. He has been trying to get the school officials to tell him.

From the school board's viewpoint, to provide such details is, in effect, to give the mayor line-by-line authority over the school system's budget, authority the board has no intention of relinquishing.

It is a problem peculiar to the school system. For example, when the mayor's budget office routinely sent operating budget forms to the Department of Transportation, the forms came back with detailed information on what was spent in fiscal 1980 and what was projected for fiscal 1981 and 1982 -- not merely how much money was to be spent for personnel, equipment and services but also how various levels of reductions would affect the delivery of services.

As a result, the Department of Transportation came back with concrete proposals: reduce the number of street lights; cut out overtime pay for handling last-minute applications for auto license plates; reduce the size of the governmental motor pool.

When the public librarian filled out the same forms, he detailed how the mayor's proposed tax cuts would mean a major reduction in operating hours and the closing of several branch libraries. Nearly all of the cuts were restored.

And so it went throughout the government.

But the school board balked, refusing even to make projections for fiscal '81 and '82 on most of the forms. And in the one case where projections were made, they were for bottom-line amounts only.

"There was no narrative explaining what they proposed to buy with the money or how it would affect the quality of education," said Gladys Mack, the mayor's budget chief. "There was nothing to indicate what savings, if any, could result from declining enrollments: just a flat figure for their overall budget.

"If they had come back to us and said our proposed cut was unacceptable because, for instance, it would require the closing of nine elementary schools, five junior highs and two high schools, or that it would mean cutting out this or that particular program, we would have been able to make some judgments."

Which, of course, is the point. The school board's considered opinion is that the mayor has no business making such judgments. That is the board's own statutory duty.

Indeed, the same statute that requires the mayor and council to set overall spending limits for the schools also requires the mayor to transmit the board's own budget request in his annual estimate of appropriations for the entire city, "with such recommendations as he deems proper."

But without some reasonable knowledge of the impact of budget cuts, the mayor's recommendations amount to nothing more than blind guesses.

But it isn't just the mayor who is operating in the blind. Those concerned parents who have been jamming the city hall chambers cannot be sure precisely what they are defending beyond a flat dollar figure.

It ought to be possible to work out some accommodation whereby the board could retain its statutory authority while at the same time detailing its plans and priorities -- if not for the mayor, then for the public.

They are, after all, our schools, and our dollars.