Born in Digne-les-Bains, Alpes-de-Haute-Provence, Alain Boghossian began playing football in the reserves of Olympique de Marseille. After a sole season on Marseille in Ligue 1, Boghossian joined Ligue 2 club Istres in order to get match practice. He returned to Marseille after a season in Istres. Boghossian moved to Italy in 1994 and was brought into Serie A club S.S.C. Napoli. He played on Napoli for three years and then played on U.C. Sampdoria for a season. Boghossian became a member of Parma A.C. in 1998, where he had his most successful years of his club career. He competed alongside compatriot Lilian Thuram on Parma. Parma won the 1998–99 UEFA Europa League, defeating Marseille 3–1 in the finals. Boghossian scored one goal at the 1998–99 UEFA Cup, the third goal in a 3–1 victory over Fenerbahçe during the second leg. While Boghossian played for Parma, the club went on to win the Italian Supercup in 1999 and the Italian Cup in 1998–99 and 2001–02. He scored the second decisive goal in the 2–1 victory of Parma over Milan in the 1999 Supercoppa Italiana final. Boghossian signed a contract with La Liga club RCD Espanyol in 2002.