The District government has lost an estimated $11 million because about 5,000 Washington parents -- most of them fathers -- have not reimbursed the city after failing to make court-ordered child-support payments.

The city agency responsible for locating delinquent parents has a backlog of more than 30,000 cases. City officials concede that they have little chance of ever collecting the millions of dollars involved.

Over the last five years, since creation of the city's Bureau of Paternity and Child Support Enforcement, the financially pinched District government has made little effort and little progress in trying to realize this revenue.

But now, taking a cue from the Reagan administration, the bureau is asking for broad new powers -- including the right to intercept income tax refunds due the delinquent parents -- to collect the money involved.

"A person should be responsible for his or her own offspring," said Eugene Brown, chief of the child support bureau, which has 10 investigators tracking down these parents and pressing cases for payment. "Some of these guys just go from one family to another, getting one woman pregnant here and another there. It's disgusting sometimes."

When parents leave home without providing money to support their children, the remaining parent usually ends up on the city's crowded welfare rolls, primarily through the Aid to Families with Dependent Children (AFDC), which is jointly financed by the federal and local governments. Each government provides about half the cost of administering AFDC, according to William Jenkins of the child support bureau.

Some absent parents are unable to make child support payments. But federal and District officials estimated that as many as half of them are able to furnish support and get the youngsters off the welfare rolls. Many of these parents don't pay, however, because they believe the chances of their being tracked down are slim.

In fact, officials said, that belief has not been far off the mark.

The city's corporation counsel, who must initiate all leagal actions for the bureau in child-support cases, has rarely had the time to become involved in the tedious work of tracking down absent parents. And the process of locating a parent and obtaining a court order for payment sometimes takes more than a year to complete.

The backlog of 30,000 cases has developed since the early 1970s when enforcement of child support was not given high priority. In the meantime, the number of single parents with children who collect public assistance has caused the AFDC program to expand dramatically.

An AFDC recipient with one child can receive $225.48 a month under the program, or $285.65 with two children.

Before a mother qualifies for the program, she must identify to city officials a child's father. The woman then becomes eligible to receive AFDC payments, and it is up to the Bureau of Paternity and Child Support Enforcement to find the father in an effort to get support payments from him.

The Reagan administration has put a high priority on forcing absent fathers to support their children.

"Over 80 percent of the people of AFDC wind up being there because the primary wage earner has either deserted the family or is not living up to his court-ordered responsibility," said Laura Genero, press secretary to Richard Schweicker, secretary of the federal Department of Health and Human Services.

The Reagan administration has proposed five measures to help cities recover support payments. One proposal, now pending in Congress, would give the Internal Revenue Service the power to intercept a federal income tax refund check due a parent who has not paid child support.

City council member John L. Ray (D-At Large) has introduced a bill in the council that would give the District the authority to intercept District tax refunds destined for people who are delinquent in support payments.