“This is not what I wanted for my son,” Ms. Himmelstein said on Thursday. “We don’t condone this.”

Jermaine, like their older son, is autistic, she said. He graduated from Washington Irving High School last June at age 20, and she proudly displayed his diploma for a visitor this week while Jermaine was out.

He is a good young man when he’s home, she said, and he does not want for money or affection.

“Jermaine comes from a home where we give him hugs,” she said.

She does not visit Times Square, and had no idea what her son was talking about when he mentioned characters he saw there. “He said, ‘I want to be like the Cowboy Ranger and the Cookie Monster,’ ” she said.

How he came to the idea of free hugs — he’s not the city’s first, not even the park’s — is a mystery to her. “Please write this down: I’m not going to stand for this for life. I’ve got him hooked up with computer school. We’ll find a way.”

Jermaine Himmelstein keeps a rigid schedule, and is always home by 8:30 p.m., so his parents worried when, on April 9, the hour came and went with no word.

Shortly before 3 p.m. that day, a female student from N.Y.U. entered the park, and Mr. Himmelstein approached with his sign.

“She says, ‘I have a boyfriend,’ and this and that,” Mr. Himmelstein recalled in the park this week. “Thinking I’m desperate, having no friends,” he became upset, he said.