Born in Arezzo, Parigi started his career in the team of the city. On 5 August 2012 he made his debut for Arezzo as a substitute replacing Filippo Borgogni in the 62nd minute of a 2–0 away defeat against Pisa in the first round of Coppa Italia. He played also 4 matches in Serie D.
After a season-long loan to the youth team with an option to buy Parigi joined to Atalanta in January 2013. He played in the youth team for 3 years before to join Cosenzaon loan.
Loan to Cosenza
On 2 February 2016, Parigi was loaned to Serie C side Cosenza on a 6-month loan deal. On 14 February he made his Serie C debut for Cosenza as a substitute replacing Giovanni Cavallaro in the 83rd minute of a 1–1 home draw against Catanzaro. On 28 February he made his second appearances, again as a substitute, replacing Andrea Arrighini in the 87th minute of a 2–1 home win over Juve Stabia. On 2 April, Parigi scored his first professional goal, as a substitute, in the 94th minute of a 3–2 away win over Monopoli. On 30 April he played his fourth match as a substitute replacing Andrea La Mantia in the 71st minute of a 3–3 away draw against Fidelis Andria. Parigi ended his loan to Cosenza with 4 appearances, all as a substitute, and 1 goal.
On 1 July 2016, Parigi was signed by Serie C side Forlì on a season-long loan deal. On 27 August he made his Serie C debut as a substitute replacing Federico Baschirotto in the 76th minute of a 1–0 away defeat against Venezia. On 1 October, Parigi played his first entire match for Forlì, a 5–0 away defeat against FeralpiSalò. After loss 3–1 on aggregate the play-out matches against Fano, Forli was relegated in Serie D, but Parigi was an unused substitute both times. He ended his season-long loan to Forlì with 20 appearances, only 4 as a starter.
On 27 July 2017, Parigi was signed by Serie C club Akragas on a season-long loan deal. On 26 August he made his Serie C debut in a 1–0 away defeat against Matera, he was replaced by Alessio Leveque. On 3 October, Parigi played his first entire match for Akragas, a 3–1 away defeat against Catanzaro. On 7 October he scored his first goal for Akragas in the 39th minute of a 2–2 home draw against Bisceglie. On 25 November he was sent off with a red card in the 30th minute of a 2–0 away defeat against Catania. On 21 January he scored his second goal in the 47th minute of a 3–2 home defeat against Monopoli. In January 2018, Parigi was re-called to Atalanta leaving Akragas with 18 appearances, 2 goals and 1 assist. On 26 January 2018, Parigi was loaned to Serie C club Virtus Francavilla on a 6-month loan deal. On 29 January he made his Serie C debut for Catania as a substitute replacing Giuseppe Madonia in the 62nd minute of a 1–0 away defeat against Catania. On 4 February, Parigi played his first match as a starter for Virtus Francavilla, a 0–0 home draw against Rende, he was replaced by Giuseppe Madonia in the 69th minute. Parigi ended his loan to Virtus Francavilla with 11 appearances, only 4 as a starter and 2 assists.
On 6 August 2018, Parigi was signed by Serie C side Paganese on a season-long loan deal. On 16 September he made his Serie C debut for Paganese in a 4–1 home defeat against Rende, he was replaced by Thomas Alberti after 64 minutes. Two weeks later, on 30 September, he scored his first goal for Paganese, as a substitute, in the 93rd minute of a 3–1 home defeat against Juve Stabia. On 20 October he scored his second goal in the 38th minute of a 4–2 home defeat against Catania.
Olbia
On 22 July 2019, he signed a 3-year contract with Olbia.
