Centre-cum-wing Odogwu was in the England squad for the entire 2021 Six Nations under former coach Eddie Jones without earning a cap.
The Coventry-born 26-year-old, who plays for Stade Francais, has now turned his attention to playing international rugby for Italy, qualifying through his father.
Harlequins' English-born No.8 Dino Lamb, Zebre props Paolo Buonfiglio and Matteo Nocera, and Toulouse's French-born halfback Martin Page-Relo are the other uncapped players named by Italy coach Kieran Crowley on Monday.
Former captain Parisse, 39, has been in good form for Toulon this season and had hoped for a recall despite retiring from international rugby following the 2019 World Cup, where he was denied a swansong after Italy's final game against New Zealand was called off due to Typhoon Hagibis.
Crowley told reporters in March that Parisse, Italy's most capped player with 142 appearances, was in his thinking for the tournament in France.
Italy will play warm-up matches against Scotland, Ireland, Romania and Japan ahead of the World Cup.
Italy training squad:
Forwards: Filippo Alongi, Paolo Buonfiglio, Pietro Ceccarelli, Simone Ferrari, Danilo Fischetti, Ivan Nemer, Matteo Nocera, Marco Riccioni, Federico Zani, Luca Bigi, Epalahame Faiva, Gianmarco Lucchesi, Marco Mandredi, Giacomo Nicotera, Niccolò Cannone, Edoardo Iachizzi, Dino Lamb, Federico Ruzza, David Sisi, Andrea Zambonin, Lorenzo Cannone, Riccardo Favretto, Toa Halafihi, Michele Lamaro, Sebastian Negri, Giovanni Pettinelli, Manuel Zuliani
Backs: Alessandro Fusco, Alessandro Garbisi, Martin Page-Relo, Stephen Varney, Tommaso Allan, Giacomo Da Re, Paolo Garbisi, Juan Ignacio Brex, Enrico Lucchin, Leonardo Marin, Tommaso Menoncello, Luca Morisi, Pierre Bruno, Ange Capuozzo, Simone Gesi, Monty Ioane, Federico Mori, Paolo Odogwu, Edoardo Padovani.
