The Supreme Court, opening its 1982 term, yesterday agreed to rule on the constitutionality of Minnesota's program of state income tax deductions for private and parochial school tuition, a case targeted by civil liberties lawyers as a test of the Reagan administration's own national tuition tax credit proposal.

The court also said it would take up cases involving the liability of cities for medical expenses of people injured by police, the authority of the federal government to recover grant money misspent by the states and the constitutionality of tax exemptions for veterans organizations that lobby Congress.

In other actions among the hundreds yesterday, the court:

* Said it would allow the Reagan administration to participate in the oral arguments in defense of one of the most controversial abortion laws in years: Akron's ordinance requiring, among other things, that a women wait 24 hours before becoming eligible for an abortion at any stage of pregnancy, and that a doctor read her a statement declaring that the fetus is a human life and describing its anatomical features in detail. The court's decision was in doubt because the government is not a party to the case, only a "friend of the court."

* Laid to final rest the ruling of a federal judge in Idaho last year that allowed states to rescind ratification of the Equal Rights Amendment. The same judge also ruled that Congress did not have the right to extend the ratification deadline. The court, without comment and without dissent, wiped out the case as moot, apparently agreeing with the government's contention that the lower court should have avoided the controversy entirely until and unless the ERA were ratified.

* Declined to consider a Boston federal court ruling that white teachers could be laid off before black teachers with less seniority as a way of maintaining court-ordered racial balance during layoffs. The case, watched closely by teachers' unions, civil rights organizations and school districts under court ordered desegregation plans, was Local 66, Boston Teachers Union vs. Boston School Committee, et al.

* Let stand an appeals court decision from California allowing a college faculty member pursuing a sex discrimination complaint access to confidential evaluations used in the decision to deny her tenure. The University of California had argued (in Regents of the University of California vs. Therese Ballet Lynn) that this information had to remain privileged to maintain the peer review system that is part of the tenure process in colleges and universities.

The court accepted 12 cases for review, about half the number ordinarily granted on opening day. The restraint apparently reflects the justices' concern, voiced in a series of speeches over the last two months, about the size of the court's caseload. In addition, the court had already accepted 126 cases last term for hearing and thus verged on being overbooked with thousands of petitions still awaiting action in the next few months.

The cases chosen by the court yesterday will probably be decided in the spring or early summer.

That delay undoubtedly will be used by opponents to stall any action on the Reagan tuition tax credit proposal pending resolution of the Minnesota case, Mueller vs. Noyes, accepted for review yesterday.

The Minnesota tuition tax program allows deductions of up to $700 from state income taxes for expenses incurred by parents of both public and private school students. The private school parents are the main beneficiaries, however, because their tuition payments alone overwhelm the minimal shop, art and other fees paid by public school parents.

The public school provision was what saved the plan from invalidation by the 8th U.S. Circuit Court of Appeals, which said inclusion of public as well as private and parochial school expenses made the law "neutral" under the First Amendment, which the Supreme Court has said bars the use of public funds designed to aid religion.

The Reagan proposal is a tax credit, also with financial limitations, which many experts consider less "neutral," and therefore more questionable constitutionally. Charles S. Sims, an American Civil Liberties Union lawyer helping to challenge the Minnesota plan, said a court ruling striking it down would probably doom the Reagan proposal. A decision upholding Minnesota, however, would not necessarily guarantee the constitutionality of the administration's proposal.

The tax exemption case (Regan vs. Taxation With Representation of Washington) stems from a challenge of federal laws which deny the lucrative exempt status to virtually all groups that lobby Congress--except veterans' organizations. The challengers, supported by groups ranging from the Girl Scouts to the National Wildlife Federation, argue that the selective exemption amounted to a denial of equal treatment under the Constitution. If upheld by the justices, Congress or the courts would have to decide whether the veterans organizations should lose their exemption or whether thousands of non-exempt groups should receive them.

The police injury case (Revere vs. Massachusetts General Hospital) comes from Revere, Mass, where police responded to a house breaking report in September, 1978, by shooting and wounding Patrick Kivlin, later charged with the crime. Massachusetts General Hospital went to court to collect its expenses for the treatment of Kivlin from the city of Revere. The Supreme Judicial Court of Massachusetts ruled that under the Constitution's Eighth Amendment, forbidding cruel and unusual punishment, the town was obligated for the expenses for "necessary medical care."

The States of New Jersey and Pennsylvania successfully contested the power of the Secretary of Education to collect on allegedly misspent funds under Title I of the Elementary and Secondary Education Act. The government appealed (Bell vs. New Jersey), saying that the case now "casts doubt" on recovery procedures of dozens of federal agencies, undercutting the "effectiveness of a principal method of controlling unauthorized expenditures."

The court also agreed to hear a reapportionment case (Brown et al vs. Thomson et al), a challenge to Wyoming's state legislative redistricting based on charges that it diluted the voting strength of one of its counties in a discriminatory fashion. A clear ruling by the justices could help the lower courts handle an onslaught of such cases stemming from the 1980 census.