The House, defying threats of a presidential veto, gave overwhelming approval yesterday to a controversial tuition tax credit bill designed to aid parents of private and parochial school pupils as well as those of college students.

Approval came on a 237-to-158 ballot after House members first voted 209 to 194 to extend the tax break to elementary and secondary school tuition as well as college tuition. The amendment had been sought by Catholic and private school groups.

The bill now goes to the Senate, which is expected to pass an even more generous version. The House also may take up later a rival plan proposed by President Carter that would extend existing federal college scholarships to middle-income students.

The measure approved yesterday would allow parents of college students to reduce their federal income taxes by up to 25 percent of the amount they spend on tuition and fees, up to a maximum $100 a student this year, $150 in 1979 and $150 in 1980.

The tax credit for private and parochial elementary and high school tuition would be somewhat less - 25 percent of tuition and fees, to a maximum of $50 a pupil this year and $100 in 1979 and 1980. In both cases, the credit would take effect next Aug. 1. The close vote on extending the tax credit to cover elementary and secondary school tuition reflected the bitter fight that has been raging between Catholic and private school groups and those representing public schools.

Spokemen for public schools, including Health, Education and Welfare Secretary Joseph A. Califano Jr., have warned that providing a tax subsidy for private school tuition would increase the exodus from public schools and seriously hurt many systems.

Liberals also have opposed extending the tax break to parochial schools on grounds that it would violate the constitutionally mandated separation of church and state. And civil rights groups have protested it would thwart desegregation efforts.

Shortly after yesterday's vote, Califano issued a statement predicting that the nation's parochial schools "will never see a dollar of" the aid the House approved in the tuition credit bill "because the courts will invalidate it." He called the measure "unconstitutional."

Although yesterday's vote was a set-back for the Carter administration, it does not guarantee the bill's survival. Carter has vowed he will veto the bill, and yesterday's tally showed there would not be enough votes to overrule him.

House Speaker Thomas P. (Tip) O'Neill Jr. (D-Mass) indicated yesterday the House probably will take up Carter's college scholarship bill later this session. Both the tuition tax credit measure and the Carter legislation are scheduled to come up in the Senate after that chamber finishes the labor law bill now on the floor.

In its action yesterday, the House rejected a proposal that would have increased the tax credit to 50 percent of tuition costs (without affecting the ceiling). And it turned down an alternative plan that would have provided taxpayers with college tuition loans in the form of tax deferrals.

The proposal to extend the tuition credit to elementary and secondary school pupils and the amendment to increase the size of the tax break both were sponsored by Rep. Charles A. Vanik (D-Ohio). The deferral plan was offered by Rep. Abner J. Midva (D-Ill.).

The tax credit bill passed yesterday would cost the Treasury $25 in fiscal 1978, rising to a maximum of $1.2 billion by fiscal 1981, compared to $1.46 billion for Carter's scholarship money plan. Under yesterday's measure, the tax credit would expire at the end of 1980.

Yesterday's vote was split sharply, even among delegations from urban, heavily Catholic districts. In all, 107 Democrats and 102 Republicans supported extending the tax credit to private and parochial schools, while 159 Democrats and 35 Republicans opposed it.