Harry Potter first saw the Thestrals at Hogwarts in September of 1995, after having witnessed the murder of Cedric Diggory the same June. J. K. Rowling explained that he hadn't seen them that June because he had not yet dealt with what he had witnessed.
Harry never saw his mother die as he was inside the cot, unable to see anything outside it, nor Quirrell, because he passed out before Quirrell's death. Harry did not see Thestrals immediately after Cedric's death in June of 1995; it took a few months before he accepted this, and then was able to see Thestrals.
"J.K. Rowling stressed that in order for someone to see thestrals, one not only has to witness death directly, but that person has to accept and understand death, and that the person will never be among the living again. Luna could see thestrals too; she witnessed her mother's accidental death and she accepted the death and understood it. So, at the time of his parents' deaths, he was entirely too young at the time to grasp what had happened."