The D.C. school system yesterday became the backdrop for the increasingly heated dispute between Mayor Marion Barry and City Council Chairman Arrington Dixon over the mayor's proposed 1983 budget.

At his regularly scheduled monthly press conference, Barry said Dixon was engaging in "political grandstanding" when the council chairman proposed to increase spending for public schools by $16.2 million while reducing spending in some other agencies by 3 percent without first conferring with the mayor or other council members.

Dixon responded gruffly. "These 3 percent cuts were prepared by me, Arrington Liggins Dixon, elected chairman of the City Council by the people of the District of Columbia," Dixon told a reporter later. He said, "I don't intend to try to reflect my colleagues . . . My colleagues will have an opportunity to review this budget in any way they want to."

Dixon said that Barry's criticism of his proposal is "out of order." He said the council is "not here to rubber stamp whatever is sent to us" but to review and assess the budget proposal.

The mayor insisted at his press conference that Dixon was playing a dangerous political game by raising false hopes that additional funds could be found for schools. He added that the chairman apparently "hasn't read it the budget carefully."

In a later interview, Barry suggested that Dixon's idea is a political ploy meant to undermine relations between the mayor's office and the school board. With his reelection campaign looming, Barry in recent weeks has attempted to patch up his strained relations with the school board.

Dixon, who earlier had raised questions about the accuracy of the mayor's revenue forecast, urged the council on Monday to increase spending for schools by $16.2 million over the mayor's proposed $285 million and reduce administrative costs in other selected agencies by 3 percent.

Dixon's higher estimates of the schools' budget needs were supported yesterday by school board President David Eaton, the mayor's pastor, and R. Calvin Lockridge, chairman of the school board's finance committee.

"Dixon doesn't have to do political grandstanding on the school budget," said Lockridge. "He's not being challenged for his seat. Really, all that this shows is that the mayor is trying to balance the school budget on the backs of the schoolchildren of the city."

The school board requested that its budget be increased from $254 million this year to $289 million next year. But that request did not include funds to cover expected increases in teachers' salaries that are now under negotiation.

The mayor originally favored reducing the school board's budget to $249.5 million, in part to reflect declining student enrollment. But under pressure, he agreed to add $14.3 million for programs and $22 million for salary increases, raising the total to $285 million.

Dixon now wants $16.2 million more for the schools, arguing that Barry's proposal is insufficient to cover the salary increases and avert program cutbacks. Eaton, the newly elected president of the school board, said yesterday he plans to ask the council for additional funds.

An analysis of Barry's school budget, prepared by Parents United for Full Funding, indicates the 1983 budget would have to total $302 million to cover expected pay increases, provide court-ordered special education classes, and keep Barry's promise not to lay off additional teachers.