Prince George's County Executive Parris Glendening yesterday proposed a $528.3 million budget for fiscal 1984 that calls for the layoff of 412 county employes, including teachers, firefighters and police officers, unless the state legislature bails him out with an extra $7.9 million.

Glendening unveiled his spending plan at a carefully orchestrated press conference in which he said that the budget figure is "not adequate to meet the county's needs." While the proposed cutbacks would adversely affect county services, he said, the reductions are necessary to achieve a balanced budget as required by law. The layoffs, if necessary, would start after July 1, when the fiscal year begins.

The budget set the tax rate 9 cents lower than last year, at $2.54 per $100 of assessed value. The reduction was necessary to keep the amount collected from property taxes below $144 million, the charter-mandated limit established by voters in 1979. Because of changes in property assessments, however, the lower rate does not necessarily mean that tax bills also will be lower.

Glendening recommended that the Board of Education receive $306.3 million, the same amount as in the current budget but $10.3 million less than the board requested. Rather than give the schools what they asked for, Glendening recommended the layoffs of 193 school personnel, including 45 classroom teachers, all 57 elementary librarians and 60 health assistants.

School spokesman Brian Porter said, "The impact of the budget on the school system will be quite severe," but he added, "Now is not the time to get into a battle of schools versus police, police versus fire. We're all in this together. We recognize that Mr. Glendening is doing his best."

Porter's conciliatory response to the proposed cutbacks was echoed by County Council members, even those who disagreed with some of Glendening's budget priorities, as well as by department heads and union leaders. The muted response was in sharp contrast to the furor last year when then-county executive Lawrence J. Hogan, a Republican, proposed an equally lean budget. County leaders said yesterday that they were resigned to the county's financial problems.

Under Glendening's proposal, which must be approved by the County Council, the 900-member police department would lose 36 officers, nearly the entire recruit class, and 27 civilian employes.

The fire department would lose 25 firefighters and nine civilians. The sheriff's department, which serves warrants and transports prisoners to and from the jail, would lose 15 deputy sheriffs, and 10 civilians.

Glendening also submitted a list of "add-backs" to be returned to the budget depending on the amount of additional revenue the county can generate, presumably through legislation passed in Annapolis. Glendening said that his order of priorities for spending additional funds will be first to eliminate layoffs in the school system and to fully fund the school board's budget, then eliminate layoffs in public safety, and finally return police and fire to their current strength.

According to Glendening's add-back list, the Board of Education would get 75 percent of the first $5 million of additional money, 6 percent of the next $5 million, 26 percent of the third $5 million, up to the board's full $10 million budget request.

Some council members said yesterday they would attempt to restore all teacher and public safety positions to the budget whether the county receives additional money from Annapolis or not. School board member Angelo Castelli said the board, which has final authority over where cuts are made, said the board would not approve any layoffs this year.

"The board is shoulder to shoulder on this--this time we're not going to cut teachers," he said.

In making reductions, Glendening said he reexamined staffing patterns since 1979, and made an effort to spare agencies that had already suffered cutbacks. In light of the school system's loss of 507 teachers last year, he said, he reduced their staffing level by 1.8 percent. Public safety departments were reduced by 4.4 percent, and other county agencies collectively lost 5.8 percent of their staffs.

The few departmental increases in Glendening's budget proposal include $1.6 million to cover an increase in the county's share of Metro operating assistance, $1 million for the replacement of high mileage police cars, $600,000 for the lease-purchase of some heavy equipment in the Department of Public Works and $441,000 for the expansion of the county's computer system.