In PoA (I think when he's first learning to repel dementors with Lupin), he does hear a bit of his father:
White fog obscured his senses....big blurred shapes were moving around him....then came a new voice, a man's voice, shouting, panicking --
'Lily, take Harry and go! It's him! Go! Run! I'll hold him off --'
The sounds of someone stumbling from a room -- a door bursting open -- a cackle of high-pitched laughter --
It's page 178 in the Brit version. And Harry only sees thestrals after Cedric's death because according to JK even if he had witnessed his parents deaths as a baby, you must be able to comprehend what you have seen. That's why Harry doesn`t see thestrals at the end of book four right after Cedric`s death. He must process it first, as a baby that would be a difficult feat.
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