The Republican takeover of the White House and Senate has strongly improved chances of tax relief for parents sending their children to private schools and colleges -- a concession that critics contend could undermine the nation's public school system.

"I think that the election is probably a mandate for tuition tax credits," said Sheldon E. Steinbach, general counsel of the American Council on Education and a member of a special educational task force appointed by President-elect Ronald Reagan.

The credits would affect the tax returns of about 12 million young people or their parents, including nearly 4 million children attending parochial schools. Legislation introduced by Sens. Daniel Patrick Moynihan (D-N.Y.) and Bob Packwood (R-Ore.) would allow them to reduce their tax payments by a maximum of $250 and would cost the U.S. Treasury between $2 billion and $3 billion in revenues.

The possibility of such aid for tuition payers is only one of a number of shifts in federal educational policy that may have been foreshadowed by the Reagan landslide.

For example, advisers to Reagan have said that they will scrap the controversial "bilingual" regulations issued Aug. 5 by the Department of Education, requiring schools to provide instruction to children in their native language until they have mastered English.

Hispanic, Asian-American and American Indian civil rights groups say the regulations are the only way to assure these children equal educational opportunities. But the regulations have been sharply attacked at the local and state level as an illegal expansion of federal authority into education.

Reagan advisers say they are not opposed in principle to programs that provide bilingual training but assert that the federal regulations, as currently drafted, illegally dictate the curriculums of local schools.Washington groups representing state and local school agencies have launched an all-out lobbying effort that might have doomed the regulations even if President Carter had been reelected.

For many of the conservatives who have been elected to the Senate, the regulations are viewed as symbolic of an expanded federal role in education, which they have vowed to end. Reagan and the Republican platform favor abolition of the Department of Education, which was set up only this year.

The shift to Republican control of the Senate is a major reason for optimism on the part of backers of tuition tax credits, an issue that has stirred passions in years past.

In 1978, the House approved the credits, but the bill failed to pass the Senate largely because of strong opposition from forces opposed to aid to parents of parochial school children. Congressional aides said last week that the new Republican majority is likely to be far more sympathetic to the measure, however.

Reagan and the party platform support the credits, and Reagan also has idicated that he is no knee-jerk backer of public schools. He wrote in his 1976 book "Call to Action" that "thousands of parochial and private schools close down because they can't compete with public schools, which drain off more and more in taxes."

Catholics were among those who switched to Reagan in large numbers on Election Day. Surveys of voters leaving polling places Tuesday showed Reagan getting 48 percent of the Catholic vote, compared with 40 percent for Carter.

The Catholic Church, which in the 1970s pressed hard for the credits in an effort to stop declining enrollments, has recently argued that parochial schools in large cities deserve support because of their large following among minorities. Some Catholics have argued that they are doing a better job of educating blacks and Hispanic-Americans in big cities than the public schools.

Currently, public school enrollment is dropping rapidly while parochial enrollment is holding steady.

One congressional aide said yesterday that he expected the new chairman of the Senate Labor and Human Resources Committee, Orrin G. Hatch (R-Utah), to hold hearings soon to review a number of options for aiding private and parochial schools and for "increasing competition and diversity without weakening the public school system."

He said that in addition to tax credits a plan to give parents vouchers that they could use to purchase education for their children at different schools probably would also be examined. However, the aide said that there would be assurances that public schools would continue to receive funds. "The idea would be to let the competition improve both systems," he said. In 1978, Hatch voted in favor of the Packwood-Moynihan proposal.