Clara, 15 at the time, pressed the call button.

[diamondcutter]

Clara Daly was sitting on an Alaska Airlines flight from Boston to Los Angeles when she heard a worried voice over the loudspeaker: “Does anyone on board know American Sign Language?”
Clara, 15 at the time, pressed the call button. An air hostess came by and explained the situation. “We have a passenger on the plane who’s blind and deaf,” she said. The passenger seemed to want something, but he was traveling alone and the air hostesses couldn’t understand what he needed.

Source: English Test, Senior High School Entrance Examination, Henan Province, 2019

I’d like to know if the phrase “at the time” modifies the subject Clara or the verb “pressed”.

 

[Tarheel]

The phrase indicates that Clara was 15 at the time (whenever that happened).

The "15" is key to that phrase. Without it it means nothing.

 

[Tdol]

It modifies 15 for me. She's no longer 15, but was at the time.