“For years I’ve mentored Asian actors to prepare themselves for the lack of Asian parts out there,” said Jose Llana, a Filipino-American actor who plays Ferdinand Marcos in “Here Lies Love.” “Until now my best strategy was auditioning for Hispanic roles as well, because I can look the part,” he added, noting that he played El Gato in the Broadway musical “Wonderland.”

Still, he has slowly started winning roles that traditionally went to white actors. He was cast as Bill Sikes in “Oliver!” last fall at Paper Mill Playhouse — a decision that surprised him so much that he asked executives if the audience would accept him in the role. (“No one batted an eye,” he recalled.) It was a far cry from his experience in the lead role of Melchior in a Sundance Institute workshop of the musical “Spring Awakening” in 2000. By the time the show reached Off Broadway and then Broadway in 2006, the role had gone to Jonathan Groff, who is white.

Trip Cullman, a director who mostly works Off Broadway, has cast several Asian-American actors, including Sue Jean Kim and Maureen Sebastian, in roles that were not explicitly Asian.

“I like it when the casting of a play reflects my experience of the world around me,” said Mr. Cullman, who is white. “Expanding my collaborators’ notions of who could be ‘right for a role’ seems to me to be a moral and political duty.”

The numbers of Asian-Americans in New York theater roles have been stubbornly low for years. Between the 2006-07 and 2012-13 theater seasons, Asian-Americans filled about 3 percent of the roles on Broadway and at major Off Broadway theaters, according to a new study by Asian American Performers Action Coalition, an advocacy group. Hispanics had roughly the same numbers of roles, while black actors had 14 percent and white actors had 79 percent.

On Broadway, where producers tend to cast Hollywood names to help sell tickets, Asian-Americans actors are still more likely to be the wise adult or the best friend. In “Matilda,” Celia Mei Rubin is in the ensemble and understudies the role of Mrs. Phelps, the kindly librarian who has been played so far by black actresses. In another musical, “Mamma Mia!,” Ashley Park is in the chorus and understudies the character of a best friend, usually played by white actress.