In the "Chamber of Secrets" Harry can't see the thestrals because he did not actually see the death of his mother. He was a newborn baby and his mother was covering his body to protect him. She was struck by Voldemort and killed then Voldemort tried to kill Harry but he was not killed. He received the "Z" shaped scar on his forehead as a result of the attack by Voldemort. He does not see them until "Harry Potter and The Order of The Phoenix." This is following "Harry Potter and the Goblet of Fire," which is the book that Harry sees Cedric die.
The ability to see the thestrals occurs only after one actually witnesses the death of another person. It is the same idea as the petrification of the people who saw the basilisk. Anyone who saw the basilisk was suppose to die immediately, however, the cat saw it's reflection in water, Colin saw it through a camera, and Hermonie saw it through a mirror. So no one actually looked right at the basilisk. J.K. Rowling is a master at "spitting hairs" in her stories. Even though Harry was present when his mom died, he did not actually "witness her death, so he couldn't see the thestrals until after he had seen Cedric die.
“Lily, take Harry and go! It’s Him! Go! Run! I’ll hold him off-”.
The sound of someone stumbling from a room-a door bursting open - a cackle of high-pitched laughter.
“Not Harry! Not Harry! Please not Harry!”.
“Stand aside, you silly girl ... stand aside, now . .
“Not Harry, please no, take me, kill me instead-Not Harry! Please ... have mercy ... have mercy”. (Rowling, 1999, pp. 134 and 178).
Hope this helps.
Harry couldn't see thestrals before he saw Cedric die because even though he was there when his parents passed away, he was only a small child then and could not exactly remember their deaths or he didn't actually see their deaths when Voldemort killed them. Harry was actually witnessing Cedric's death at an age where he can actually remember it.
Harry never actually saw his mother die. She covered his face and protected him from Voldemort as Voldemort killed her. Since he was shielded from seeing her, he was unable to actually see her death. Obviously he would be able to see thestrals after Cedric dies because he sees, plain as day, Wormtail killing Cedric.
I may be a year late on this question, but that is something I always found curious. I think it's one of the Potter series' unexplained mysteries. Baby Harry did see Lily die, we can assume, as she was standing in front of him as a human shield. Unless he looked away, he would have seen his mother die. He even has flashbacks of her screams and the bright green light. One possibility is that he was not old enough to comprehend death. Possibly one cannot see thestrals unless they understand that what they witness is a death.
Or, perhaps Ms. Rowling just made a mistake. Hard to believe, but possible. :)
I am curious as to whether professor Quirrell counts as a person? Harry killed him and was looking right into his face when he died. Well I suppose it is assumed that he dies, but still would this count?