Born in Genoa into a family of sculptors, Parodi developed his facility with wood, then transferred his mastery to marble in the 1670s. His two extended sojourns in Rome refined his style; he joined the studio of Bernini as an assistant (1655–1661), although he appears to have been influenced by Algardi and his pupil Ercole Ferrata. Later on returning to Genoa, he met the French Baroque sculptor Pierre Puget, who stayed in Genoa from 1661-1666. Parodi developed a large studio to handle a large number of commissions.
In Genoa during the 1661-1670s, he completed an Ecstasy of Saint Martha for Santa Marta, a Saint John for Santa Maria di Carignano, and a Virgin and Child for San Carlo.
In 1691 he was called to Padua, where he and his studio were responsible for the six white marble sculptures of saints and the Glory of Saint Anthony (1689–97) in the polychrome marble setting of the Cappella del Tesoro ("Chapel of the Treasure") at the Basilica of Saint Anthony of Padua. The cornice is crowded with celebrative angels by a stuccador from Lugano, Pietro Roncaioli.
For Johann Adam Andreas I von Liechtenstein of Vienna, he produced two allegorical busts: Vice and Virtue, which remain in the Liechtenstein collection, Vienna. The expressive bust of Vice has a specific Bernini source in Bernini's Anima dannata.
His sculptures commissioned by Eugenio Durazzo in 1679 during the renovation of the Palazzo Balbi Durazzo, Genoa, remain in situ (the present Palazzo Reale); they are a sentimental Christ at the Column for the chapel and a set of four mythological figures from Ovid's Metamorphoses (Venus, Clytie, Adonis, and Hyacinth) for the garden. The statues are emotive and often witty reworkings of sculptures by Bernini.
Parodi's name is often invoked in connection with carved and gilded Late Baroque Genoese side tables and gueridons. The reason for this is because of his association with an assistant in his atelier, the young Andrea Brustolon. In the first part of his career, Brustolon is documented at several Venetian churches where he executed decorative carving in such profusion that he must have quickly assembled a large studio of assistants. As with his contemporary in London, Grinling Gibbons, almost all the high quality robust Baroque carving in Venice has been attributed to Brustolon at one time or another. His furniture included armchairs with figural sculptures that take the place of front legs and armrest supports, inspired by his experience of Bernini's Cathedra Petri. The gueridon, a tall stand for a candelabrum, offered Brustolon unhampered possibilities for variations of the idea of a caryatid or atlas: the familiar Baroque painted and ebonized blackamoor gueridons, endlessly reproduced since the eighteenth century, found their models in Brustolon's work.
Parodi also worked with Giacomo Antonio Ponsonelli (1654–1735) an Italian late-Baroque sculptor who was also his son-in-law. Parodi's son, Domenico Parodi (1672–1742), was a painter of some merit, initially apprenticed with Sebastiano Bombelli, then, in the early 1690s, working in the studios of Carlo Maratta and then his pupil Paolo Girolamo Piola. Other pupils of Parodi were Angelo de' Rossi, the brothers Francesco and Bernardo Schiaffino.
Other works 
- Glory of the Magdalene, high altar, S. Maria delle Vigne, Genoa .
- Boy with a Skull (Vanitas) Hermitage Museum, Saint Petersburg
- Winter from a set of Seasons (Cleveland Museum of Art)
- Rudolf Wittkower, Joseph Connors, Jennifer Montagu, Art and Architecture in Italy, 1600-1750, 1999, Part 3 p. 63.
- Christ at the Column
- Bruce Boucher (1998). In Thames & Hudson. Italian Baroque Sculpture. p. 78.
- Wittkower, Rudolf (1993). Pelican History of Art, Art and Architecture Italy, 1600-1750. 1980. Penguin Books Ltd. p. 448.