Harry Potter Series - Harry Already Saw Someone Die
In Book 5, we learn that the thestrals can only be seen by a person who has seen someone else die. Harry was able to see them in "Order of the Phoenix" because he saw Cedric die in "Goblet of Fire". However, Harry was present when his parents were killed when he was 1, so he should have been able to see the thestrals through the whole series.
Mrs. Rowling said that the death needs to have "sunk in" as Harry was a baby when his parents died he would not have remembered it, ergo those deathes have "sunk in."
In book 7 of the series, it describes (through a series of flashbacks from Voldermort's point of view) the night of Lily and James' death and states that Harry had been put in his cot when his mother was begging for his life. The chances of him actually seeing Voldermort kill her are slim. Combine this with the fact that all he can remember of that night is a green flash of light and voices, no other images and it is possible to conclude that Harry never actually SAW his parents die.
This is one of the most frequently repeated bogus "slip-ups" of this series.
Harry was in his cot when his parents were killed, so he didn't see it happen. Even if he had, he was only 1 so wouldn't have understood what he saw.
Harry didn't see Professor Quirrel die (in Book 1), he was already unconscious when this happened.
The first death he actually saw was Cedric Diggory's, and his going home for the summer break happened only two weeks later, when he was still in the denial phase.
Thus, the earliest he could have seen the Thestrals was at the start of Book 5.