The Amazing Spider-Man never had the chance to get to know his parents before their deaths sent him on a path towards becoming a superhero. Well, until they suddenly returned alive and well, of course.
Though Peter Parker was raised by his elderly Aunt May and Uncle Ben, his biological parents, Richard and Mary Parker were far from ordinary. Richard Parker served as a special forces operative, before he was recruited to be an intelligence officer, while Mary Fitzpatrick, herself the daughter of an O.S.S. operative, worked at the same organization Richard did. The two quickly fell in love, and after Richard and Mary eventually married, they worked together on many undercover operations (even rescuing Wolverine at one point). Richard and Mary continued their undercover operations after having Peter, often leaving their son in the care of Ben and May. But the Parkers' luck would run out, as a mission to infiltrate the Red Skull's organization went awry, and the villain had them killed in a plane crash. But that wasn't the end of the intrigue.
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In 1965's Amazing Spider-Man Annual #5, Peter discovered his parents, in the eyes of the United States Government, were considered traitors. Unbeknownst to the world at large, the Red Skull had planted false evidence on Richard and Mary Parker, meant to sow doubt as to their real allegiance. When their bodies were discovered, the government had to officially acknowledge both Richard and Mary as traitors. Refusing to believe that his parents were actually turncoats, Peter traveled to the location of their last mission in Algeriato prove their innocence. Spider-Man came into conflict with the Red Skull and, after a long battle, found the evidence in the Skull's possession that determined that his parents were innocent all along. Peter had restored his parents honor, and they were considered heroes once again. But the story of Peter's parents was far from over.
In 1992's Amazing Spider-Man #365, Peter's parents had seemingly returned from the dead, making their presence known to Peter, Aunt May, and Peter's wife Mary Jane. The following issue explained that the Soviet Union had gotten involved in Richard and Mary's mission with the Skull, and that the bodies found on the aircraft belonged to soviet soldiers, rather than Richard and Mary. Though the two were presumed dead, Richard and Mary were detained in various Soviet prisons for many years, until the Soviet Union dissolved, at which point the two were freed. Though Peter is initially hesitant to accept these two as his parents, he manages to confirm their story after another encounter with the Red Skull, and gradually warms up to them. Eventually Peter even reveals his secret identity to his parents... at which point their true nature is revealed.
As it turns out, Peter's parents had actually died in the plane crash, and that the "Richard and Mary Parker" he had come to know were robots created by longtime Spider-Man villain The Chameleon. The Chameleon sought revenge against the Wall-Crawler for the death of Kraven the Hunter, his half brother, during the story "Kraven's Last Hunt." Chameleon had determined that Peter Parker and Spider-Man were friends, and created both "Richard and Mary" to act like Peter's real parents in order to figure out if Peter Parker knew who Spider-Man really was. Once "Richard and Mary" learned the truth, they returned to the Chameleon to share Spider-Man's secret, though Spider-Man himself arrived and managed to stop his fake parents. In the aftermath, the Chameleon manages to escape -- but both "Richard and Mary" were left dead, leaving a broken Peter to anguish at the idea that he had truly lost his biological parents (all over again).
Though this story-line only lasted two years real time, it had major consequences for Spider-Man moving forward. Spider-Man spent several issues hunting down the Chameleon for his role in this plot, and learned that the true mastermind behind Peter's parents reemerging was Harry Osborn, the recent Green Goblin at the time. Though Harry had died a year prior, this plan from beyond the grave drove Peter over the edge, causing him to renounce Peter Parker, and only be Spider-Man. This drastic change to the character ultimately led to the infamous Clone Saga, an attempt to bring Spider-Man back to basics without any of the baggage of the last 30 years. Though things would eventually return to normal for Peter Parker, it is interesting to note that a drastic change to the status quo, the revival of Peter's parents, would inevitably lead to even more changes to the character.
Richard and Mary Parker have had an interesting history in the Marvel Universe, despite their deaths occurring prior to Amazing Fantasy #15. Richard and Mary have also undergone many changes over the years as well, appearing in the Ultimate Universe as scientists, rather than secret agents, responsible for that universe's version of Venom. This incarnation of Peter's parents was used in The Amazing Spider-Man films, with the two working for Oscorp before going on the run to protect Peter from the evil organization. Even though Richard and Mary don't get mentioned often, recent developments in the Spider-Man comics have introduced Peter Parker's biological sister, Teresa, making Richard and Mary relevant to current Spider-Man continuity.
Though Richard and Mary may be Peter's biological parents, Aunt May and Uncle Ben were responsible for raising and instilling responsibility in Peter Parker, and without them, Peter may not have turned into the hero we know and love. Still, despite not knowing his parents, for a brief moment in continuity Spider-Man got as close as he could... even if it was a cruel ploy.