LOS ANGELES — Bill Delaney’s two little girls spend three nights a week with their fathers, at the home Mr. Delaney shares with his husband in San Francisco. The other nights, they stay with their mothers, a lesbian couple who live nearby.

The girls have four parents — a result of a kind of nontraditional family arrangement that has become increasingly common. But officially, California, like most other states, recognizes no more than two legal parents.

That limit could soon be lifted.

A bill moving through the California Legislature would allow judges to recognize more than two legal parents for a given child, opening the door for alternative families to seek legal recognition of their relationships.

“There are literally scores of different families and circumstances,” Mark Leno, the state senator who sponsored the bill, said.