Piastri, last year's winner from pole position in a McLaren one-two at the Circuit de Catalunya, finished behind George Russell, who was looking strong on Friday after two races without scoring.
The Mercedes driver was placed just behind fellow Briton Norris in second practice after topping the list on practice one.
Earlier in the day, Piastri had finished second fastest in the first free practice session with Ferrari's Charles Leclerc third.
Mercedes' championship leader Kimi Antonelli, who like Norris sat out the first session while the team gave track time to reserve driver Frederik Vesti, was placed fifth.
Red Bull's four-times world champion Max Verstappen was sixth fastest, with Racing Bulls' Arvid Lindblad seventh and Audi's Gabriel Bortoleto eighth.
Ferrari's Lewis Hamilton, Antonelli's closest championship rival but still 66 points behind after finishing runner-up in the last two rounds, was ninth.
Russell enjoyed a trouble-free first session as he lapped with a best effort of 1:16.363, 0.203 quicker than Piastri.
First practice saw a cast of reserves on track with McLaren's Leonardo Fornaroli fifth in Norris's car.
Others in action included Paul Aron in an impressive sixth for Audi, Dino Beganovic at Ferrari, Ayumu Iwasa at Red Bull and American Colton Herta, a nine-time winner in IndyCar, at Cadillac.
Luke Browning had been due to take Alex Albon's Williams for the session but failed to set a lap time due to an electrical issue.
Barcelona is hosting the seventh round of the season, with Antonelli chasing a sixth win in a row.
He has made F1 history on several counts this season. At age 19, he became the youngest pole-sitter en route to his first win in China, followed by becoming F1's youngest points leader after a win in Japan.
The bushy-haired Italian just kept going, sweeping the alliterative triple of Miami, Montreal and Monaco to make it five in a row and tie the longest winning streak ever managed by F1 victory leader Hamilton.
