Mr Allen was wanted by police after the bodies of Philip Howard, 44, Janet Jordan, 48 and six-year-old Derin were found two and a half hours later at their home in Didcot, Oxfordshire on Saturday evening.
On Monday more than 100 police officers were searching parks on Oxford where Mr Allen, aged 21, had been employed as a groundsman.
Police believe 21-year-old Mr Allen arrived in Oxford from Didcot at 5.45pm on Saturday.
Detective Superintendent Chris Ward, the head of Thames Valley Police’s major crime unit, said: “Jed is wearing a red or maroon shirt, a dark hooded jacket with a white trim on the zip, blue jeans and black shoes.
“After visiting WH Smith, we believe that Jed then walked to Oxford University Parks.”
More than 100 police officers are searching the parks, where Mr Allen worked as a groundsman.