I was wondering the exact same thing. I'm guessing that Harry could only see the Threstals in his fifth year, because his parents' death happened when he was way too young to remember, and the event did not really affect him until he was much older. Sure, he wished his parents were alive, but the their death was not traumatizing as he could only remember a flash of green light. Cedric's death was at an age where he could really understand what was going on, and the impact of it at the time.
He was sleeping.
Rhr4eva's answer: I just thought I would add that no, in fact he wasn't. In the 7th book they had some sort of memory of Voldemorts that went into Harry's head becuase of the connection, and it said that Harry looked like he was waiting for his mom to jump up and laugh, and for his father to take off the "mask" that was Voldemort. Something like that, anyway. So Harry wasn't asleep. He just didn't know what was going on, I guess.
One can see a Thestral when one has seen Death. In order to 'see' death, it is implied that one must be able to comprehend and accept it. As an infant, he couldn't comprehend it, in his first year when Quirrel died, he passed out before it happened, and in his fourth year, it took it a while for Cedric's death to 'sink in' and be fully accepted.. hence why he didn't start to see Thestrals immediately, but only after a few months.