Which of these two people killed this spy family?

Jared killed his own parents because he *wanted* to be a spy for the Russians, and they were trying to discourage him.

At different points in the second season, different parties had varied other motivations, including Larrick and Claudia. It turned out it wasn’t either of them. The killer was the Connors’ own son, Jared, who had been groomed as a second-generation illegal behind his parents’ backs (by Kate - not Claudia), and decided to homicidally fight them when they objected to his attempt to follow in their stealth footsteps.

It was Kate who converted Jared to the cause, at the Centre’s insistence, without parental permission. This also explains more clearly why Claudia got booted from her post and Kate was installed in her job. The Centre knew that the best way to win over a teenage boy was by assigning a young, pretty blonde to do the persuading.

To answer for Claudia's comments, she was trying to manufacture a believeable scenario, because she knows that Elizabeth's children are future prospective Russian spies, and she is unsure of their devotion to Russia to also similarly disallow it. She had to have a cover for the recruitment, which was the true motivation behind the killings. She was also trying to look good to retain her job, by taking credit.