With the TPC-Four Seasons course vulnerable due to an unusual lack of wind, Palmer equaled his lowest score of the year in his 40th round of 2012.
Marc Leishman and Alex Cejka both shot a 65 while the group at 66 included Matt Kuchar, who last week won The Players Championship. Scott Piercy, Charley Hoffman, Bill Lunde, Blake Adams, Dicky Pride and Andres Gonzales shared fourth place with Kuchar.
Kuchar is trying to become the second two-time winner on the tour this year. He would join Hunter Mahan, who is not competing this week.
Defending champion Keegan Bradley, who followed up his victory here a year ago with a major title at the PGA Championship, was part of the large crowd at 67. Bradley overcame a slow start by going birdie-birdie-eagle beginning at the 14th.