Thirteen times since 1966 a tuition tax-credit bill has gained life on Capitol Hill, and 13 times it has died.

Two weeks ago President Reagan sent Congress a 14th version.

But though the president has committed himself to the bill and has risked a political defeat, doubt remains on Capitol Hill as to whether he is serious or simply playing to conservatives and Roman Catholic school parents, who have been pushing the bill for the last 20 years.

Reagan appeared serious two weeks ago when he gathered Sens. Robert J. Dole (R-Kan.), Russell B. Long (D-La.), Daniel Patrick Moynihan (D-N.Y.) and Alfonse M. D'Amato (R-N.Y.), a group of congressmen, Attorney General William French Smith, Treasury Secretary Donald T. Regan and Education Secretary Terrel H. Bell and told them to give the bill "the highest priority."

The serious treatment continued that afternoon as White House chief of staff James A. Baker III met with some of the bill's strongest supporters: the Moral Majority, the U.S. Catholic Conference, the Council for American Private Education, the Knights of Columbus, Agudath Israel of America and Citizens for Educational Freedom.

Repeating the president's message, Baker told the gathering that tuition tax credits are a White House priority this year, the tax credits are in the budget, the president spoke up for the measure in his State of the Union address and the White House will do its part and twist arms if the groups will do their part and put pressure on Congress.

""We told the legislative office that when they finish with the budget and Social Security the first thing on the plate is tuition tax credits," said one White House official.

But no matter how strong the president's backing, the word on Capitol Hill is that the bill is going nowhere again this year.

Rep. Dan Rostenkowski (D.-Ill), chairman of the House Ways and Means Committee, told reporters that while he still supports the idea, the budget deficit is too large for Congress to be considering additional new tax credits.

Aides of senators who support tuition tax credits say that the bill is a "swamp water" of racial questions, budget questions and angry public school supporters. And, while the sentiment that saw the bill pass the House and the Senate Finance Committee last year remains high in Congress, the additional 26 Democrats elected to the House last fall may have tipped the scale against the measure this year.

Adding to the the bill's problems are questions of whether it is another version of the Bob Jones University case, in which the administration sought to give tax exemptions to segregated schools.

Further complications could arise next month, when the Supreme Court hears arguments on the constitutionality of state-funded tuition tax credits in Minnesota.

The largest bone of contention is whether public dollars should be used to subsidize parochial and exclusive private schools. Adding to the difficulty is the fact that some schools are based upon religious affiliation, and government subsidies could violate the First Amendment's separation of church and state.

The tuition tax credit plan is the first of three White House proposals for aiding parents of students. The second would let low- and middle-income families establish education savings accounts with tax-free interest. The third would aid parents of disadvantaged students by giving them vouchers for money that now goes directly to public schools.

Political aides in the White House hope that the proposals will rebuild Reagan's support among the working class, single lower-income women and people who believe the administration favors the rich.

To blunt criticism that tuition tax credits are a subsidy for wealthy children attending private schools, the White House is selling the idea as a tax break for working- and middle-class parents and has titled the bill the "Educational Opportunity and Equity Act of 1983."

In the letter that he sent to Congress with the bill, the president also said that tuition tax credits would reduce attendance at public schools and decrease the burden on local and state taxpayers.

Under the plan, tax credits could cover up to 50 percent of tuition. The plan would phase in over three years, starting this year with a maximum $100 deduction that would rise to a $200 deduction in 1984 and a $300 deduction in 1985.

The full credit would be available to parents with adjusted gross incomes of up to $40,000, and smaller credits would be available for parents with incomes of up to $60,000.

The plan would cost the government $200 million in 1984, and this would increase to $800 million by fiscal 1986, the White House says.

Last year the president proposed a maximum credit of $500. That was reduced to $300, and the plan cleared the House and the Senate Finance Committee before it died.

White House aides say that the new bill's phased-in credits and anti-discrimination language should make it more appealing to legislators concerned about the racial and budget implications of tuition tax credits.

Two provisions that had divided the bill's supporters have been deleted. One required schools to take handicapped students, a provision that had been opposed by Catholic schools lacking the necessary and costly facilities. The second, which made school attendance compulsory, was opposed by fundamentalist schools as interfering with a parent's right to decide when and where to educate a child.

Supporters of public schools, including the national Parent-Teacher Association, teachers' unions and civil rights groups, argue that the tax credit would damage the nation's system of public education further.

"How can the adminstration justify federal aid for private education at a time when it is also asking for spending cuts that will seriously harm public schools?" asked Albert Shanker, head of the American Federation of Teachers.

"This [tuition tax credits] is bad public policy no matter how you cut it," said Willard McGuire, president of the National Education Association. "This nation agrees to bear the obligation of free universal public education for all its children. There is agreement to the right of private schools as an alternative if the individual wants it and wishes to pay for it. But there is no agreement that the public should have to pay for that. Our obligation is to the best possible public school system."

Yet the White House appears to be standing firm behind the tuition tax credit bill and to be convinced that it will attract constituencies as diverse as single women with children, the working class, Jews, Catholics and conservatives.

"This is not school prayer," said Ron Godwin, executive vice president of Moral Majority. "This one is more than a conservative issue. It has a large following of parents across the board. This one will be with us a long time, because it is an economic thing, and politically it has appeal to parents."

Rabbi Menachem Lubinski, director of government and public affairs for Agudath Israel of America, the major orthodox Jewish group, said the bill has wide support among liberals as well as conservatives.

"This is not necessarily a conservative issue," he said. "There is a broader constituency. The Catholic schools and Jewish schools, the growing number of Jewish day schools, the fundamentalist schools--there is a whole middle-American body that wants this, and I think the White House knows it."

Proponents of the bill argue that there is no church-state constitutional violation inherent in tuition tax credits or even a problem with the use of public dollars to support private schools.

"The money does not go to institutions but to individuals," said Larry Uzzell, of Learn Inc., a conservative think-tank. "It is the same as a general student loan."

But the principal question about tuition tax credits is the depth of the president's commitment.

"My impression is, Ronald Reagan is a true believer in this," said Willis D. Gradison Jr. (R.-Ohio), who attended the White House meeting at which Reagan introduced the bill.

"He didn't give us some formal statement about tuition tax credits," Gradison said. "He stayed and talked at length about an independent school on the South Side of Chicago [Providence-St. Mel's, a mostly black, parochial school].

"He said a remarkable man runs the school and he [Reagan] has tried to raise money for them. He seemed to really warm to the subject. He talked about how they stress discipline, how it was in a neighborhood where most children have no hope of educational advancement . . . .