Harry had already witnessed his parents'death. Why was he not able to see the Thestrals (resulting in self-driven carriages) till he witnessed Diggory's death? Also, after he did so, at the end of the fourth part, he still is being carried in a self-driven carriage (at least in the book)?
He may not have witnessed his parent's death. He was a baby, might have been asleep (despite the movies' description where he is directly staring at Voldemort). Another option is that you must remember these deaths, or have their psychological trauma. Since Harry was a baby at the time, he only remembers hearing his parents, and the green light from Voldy's Avada Kedavra at him.
At the end of the fourth part, he might simply not notice the Thestrals pulling the carriages. He had a lot on his mind then, and the books show us his PoV.