The neighborhood day-care center may be replacing the little red schoolhouse in the national psyche, but about half of the nation's young children with full-time working mothers still are cared for by a relative or parent.

Another 15 percent are enrolled in day-care centers, and most of the rest spend much of their days in what are known as family day-care homes, where people, mostly women, take small groups of children into their homes for a fee.

Family day-care homes are a mammoth, largely unlicensed and uncounted cottage industry. Children's advocacy groups and government officials only can guess at their number. The homes typically are about $10 to $30 cheaper a week than day-care centers, and they tend to have more flexible hours and accept children at a younger age. Some are glorified baby sitters; others have had training. In most cases, it is the price tag or the small scale that attracts parents.

"It's a conscious choice by most parents," said Joe Perreault of the Save the Children Federation. "They'd rather have their kids in a house," he added, because they believe it means more personalized care and perhaps a healthier environment.

There are about 2,000 licensed or certified family day-care homes in the Washington area, according to a study by the Greater Washington Research Center. Local officials estimated that nine times that many may be in operation but are not registered.

Although most states require family homes to be licensed, most are not. This troubles children's advocacy groups, whose members all can tell horror stories of babies and toddlers crammed into small rooms with blaring televisions.

They worry many of the providers are operating homes that are not safe for children and have little idea of how to stimulate and challenge children. The same thing could be said of some full-time mothers, of course, but day-care specialists say that when homes are licensed the quality of care has been shown to improve.

Because the licensing effort has failed, many states, including Maryland, have turned to registering homes in the hope that more will come forward to be counted and judged. But not everyone is convinced that registration will be more successful.

"It's not fear of licensing that keeps people back," said Kathleen Murray of the Child Care Law Center in San Francisco. "It's ignorance or neighborhood restrictions against operating a business in your home."

The fact that the government cannot tax a business that it cannot see also may have contributed to the failure of licensing. One lure that has turned up significant numbers of homes is the Agriculture Department's day-care food program.

"They come forward to claim the money," one official said. "And when they do, they wind up asking a lot of questions and learn how to improve their operations, and the care gets better."

Because the program doles out food to all children, regardless of family income, the Reagan administration has proposed eliminating it.

Recently, around Washington and elsewhere, day-care homes have begun to cluster into what are called day-care systems. In these systems, a few homes are screened by a central office, which also solicits business and refers families to the homes. About a dozen such systems are in operation in the Washington area.

"Some of them have not been able to supply enough children to the homes to make the system worthwhile," one local official said. "If they can't provide them, then the system tends to fall apart."