I do not know whether or not the parents had a key to Julia's apartment. I'm sure if LE does not know the answer, this may be a question for JT and KT.
Thank you, Trojan. I had thought the answer wasn't given and questions about the key come up since it was reported as missing.
Using the thought it was the apartment key, the killer could have taken it knowing they would access to the place later because he would have knowledge when it was empty. Perhaps to go back to see if they missed anything or see what was taken out....
So two 'why' questions:
Why would the killer want that key?
Now using the thought that perhaps the parents might have had a key, why would JT need to take hers if he already had one?