From IMDb
While the on-screen director's credit goes to Tobe Hooper, there is a wealth of evidence to indicate that writer/producer Steven Spielberg was responsible for most of the creative decisions in the movie, with Hooper merely responsible for the mechanics of shooting scenes.
Steven Spielberg worked on this movie and E.T. the Extra-Terrestrial (1982) literally back to back. Principal photography on Poltergeist ended in August of 1981, then Spielberg took a few weeks off and began work on E.T. Spielberg also supervised the visual effects for both films simultaneously (which were produced at Industrial Light & Magic under the supervision of Richard Edlund and Dennis Muren). Once post production work on Poltergeist began in early 1982, Spielberg was in total control. He was responsible for the editing of the film (Spielberg's usual editor Michael Khan edited this film while Carol Littleton edited E.T), the final sound mixes and loops, the supervision of the visual effects, and the selection of Jerry Goldsmith as the composer of the score. Poltergeist and E.T opened to theaters nationwide only a week between each other during the summer of 1982, Poltergeist on June 4th and E.T. one week later on June 11th.
Steven Spielberg had a major hand in the production of this film at the same time that he was directing E.T. the Extra-Terrestrial (1982). He later said "If E.T. was a whisper, Poltergeist was a scream".