The senior All-American wide receiver, who has been sidelined all season with a broken bone in his foot, could play on Saturday against North Carolina State in Papa John's Cardinal Stadium at 3:30 p.m.
"He's close," U of L coach Bobby Petrino said. "He had a really good weekend then he practiced (on Tuesday), I'm encouraged. But I was encouraged last Tuesday, too.
"But he did look good in practice yesterday."
The 6-foot-3 totaled 55 receptions for a team-high 885 yards and tied a school record with 12 touchdowns last season. During the summer, Parker was named to the Biletnikoff and Maxwell Awards preseason watch lists.
Parker had been considered one of the top wide receiver prospects in the country.
"He's a dynamic player," Parker said. "He has the ability to get open, catch the ball and run after the catch. It's something that we've missed."
Petrino also confirmed Will Gardner will start at quarterback after going 10 of 16 for 150 yards and one TD in the second half against Clemson in his return from injury.
"Going back and evaluating everything on Saturday, we decided he should be the starter," Petrino said.