Their bodies were discovered in a house in Vicarage Road, Didcot, Oxfordshire, after officers were called by a member of the public on Saturday evening at approximately 8.20pm.
It's believed all three were stabbed to death.
Armed officers are involved in the search for the suspect, although there is no suggestion that he is carrying a weapon.
Speaking at a press briefing, Detective Superintendent Chris Ward from Thames Valley Police's Major Crime Unit described the incident as a "tragic and sad event", offering his condolences to the family.
Mr Ward said: "In relation to the search for the suspect, we are still carrying out a number of searches. Some of those searches are with armed officers, there is a very small number of armed officers who are assisting with that search.
"That's purely for the safety of the searching officers."
He added there was "no suggestion whatsoever that that person has a weapon and certainly has a firearm".
"So where members of the public see officers with shields, that's purely in relation to the protection of officers carrying out the searches."
Allen is described as a white male, 6ft tall, and of large build.
"It is very important that we locate him as quickly as we can," Mr Ward said.
As news of the tragedy began to spread, people could be seen visiting the scene and leaving floral tributes.
A yellow teddy bear was also placed just inside the cordon, and a hand-written tribute was attached to a bunch of flowers.
Part of it read: "To my best friend, my soul mate, Jan, Dezzy and my anorexic Phil and my gorgeous Jed.
"You are my family and always will be. I will miss you all each day... You have all left a massive void in my heart which will never be filled.
"I love you all so much, Jan, Phil and my Dezzy Doo."
Although Allen - who is known to police - is not considered a danger to the wider community, the public should not approach him and instead contact police.
Officers received the tip-off at around 8.23pm and detectives are keen to hear from witnesses who may have been in the area of Vicarage Road at the time.