Former world number one Woods, a 14-time major winner in his first US PGA event since last August's PGA Championship due to a back injury, opened with a two-over 73 Thursday and was three-over through five holes in his second round Friday.
Woods, who began on the back nine, made a bogey at the 11th hole and a double bogey on the 14th and fell five strokes off the projected cut line.
Palmer opened with a seven-under 64 and remained atop the leaderboard after the final nine players who did not finish Thursday due to darkness completed round one Friday.
His big threat, fellow American Daniel Berger, made pars on his final two holes to finish on 65.
This enabled the rookie to share second with 2011 PGA Championship winner Keegan Bradley and Bubba Watson, the reigning Masters champion.