The City Council Wednesday ripped into the de Blasio administration for rejecting a proposal to offer free lunch to every child in public school.

The spat is a rare conflict at City Hall, pitting Mayor Bill de Blasio against the council and the city's public advocate, Letitia James, all usually allies who rarely criticize one another.

Children from lower-income families currently get free lunch, but proponents of the free-lunch idea want to make the benefit universal, regardless of wealth or poverty, to remove the stigma of being a handout recipient.

"You just don't get it," James, an ex officio member of the council, told schools Chancellor Carmen Fariña, during a hearing on the education budget.

The disagreement centered on whether the proposal, which would cost the city about $24 million, would jeopardize federal funding distribution for poor kids. The administration says yes. The council says no. Fariña says the administration is exploring the issue.

Before Fariña testified, dozens of student athletes from an underfunded athletic program attempted to present her with a petition demanding equity in funding. When their league founder began shouting, the group was escorted out.

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The council and de Blasio are also sparring over the NYPD's head count, now about 35,000, and whether to increase it by 1,000 officers. The council favors the hike; the administration does not.

Negotiations of the $73.9 billion budget must be complete by July 1, for the beginning of the upcoming fiscal year 2015.

Later Wednesday, council members grilled new parks department Commissioner Mitchell Silver -- who has 16 days on the job -- about diverting resources to smaller parks that cannot raise private funds as well as, for example, Central Park, which is managed by a nonprofit conservancy.

Council members, including Mark Levine, (D-Manhattan), chair of the council's parks committee, want $27.5 million in extra funding toward "parks equity" to hire more gardeners, maintenance workers and park police. Those funds are not in de Blasio's executive budget, though the mayor has said he supports the concept of parks equity.

Silver, formerly the chief planning and development officer for Raleigh, North Carolina, said "maintaining and improving small parks in less wealthy neighborhoods" is a "worthy goal," but added that his department is still assessing how it might go about redistributing resources.

"We don't know 'the how' at this point," he told reporters after the hearing. "Certainly, we're going to look at the existing capital budget, but we're also going to have a conversation with the conservancies."