Jess Parker's flat was the home of Anomaly Research Centre field co-ordinator; Jess Parker. The flat was personalized to Jess's "standards", like being full of Feng Shui and many vintage items, such as multiple old telephones.
History[]
After Connor Temple and Abby Maitland return to the present after a year in the Cretaceous, Jess invited them to stay at her flat until they could find their own place. They presumably stayed with Jess as they never mentioned moving out or actually buying their own flat. (Episode 4.2)