Spider-Man (Peter Parker) is an exceptional science prodigy, who was bitten by a radioactive spider on a class field trip, which turned him into Spider-Man.
Since Sony and Marvel made a deal such that Spider-Man is now part of Earth-199999 (the Marvel Cinematic Universe), and is going to appear in Captain America: Civil War, and his new solo movie in 2017, Marvel's Spider-Man, with the hero being played by Tom Holland. In this version, he lives in New York along with his Aunt May, played by Marisa Tomei.
Though Skye clearly parodies his creed, Spider-Man exists in the MCU reality. Skye's line is a nod to him.
Though there is no way of knowing if this will be the case in the series, in the comics, Spider-Man's parents Richard and Mary Parker were S.H.I.E.L.D. agents.
Spider-Man also happens to be an agent of S.H.I.E.L.D. (along with Nova, Power Man, Iron Fist, and White Tiger) in the animated series "Ultimate Spider-Man".