He learns in Captain America: The Winter Soldier.
When Captain America and Black Widow find the hidden SHIELD bunker that contains Arnim Zola's AI, one of the things that Zola explains is how HYDRA infiltrated SHIELD from the beginning. Rogers asks how that's possible and how no one found out, and Zola replies that over the years, "accidents have been made to happen".
His video screen then flashes through a series of "accidents" that he's basically admitted to being arranged by HYDRA, and the first image in that of Howard Stark.
Earlier in the movie, Black Widow had revealed that The Winter Soldier had been killing people for ~50 years.
Once Cap discovered Bucky's identity and that he's working for HYDRA, and that it was Bucky who arranged Nick Fury's "accident", it wouldn't have been much of a leap at all to figure out that he's also the one that assassinated the others Zola bragged about. (Note that when Tony first confronts him, he actually say something like "I didn't know it was him", but doesn't push the issue, implying that he didn't know for sure that Bucky was the assassin, but had come to that conclusion anyway.)