A's manager Bob Melvin said Friday that Parker has right forearm tightness and soreness and would be examined Monday by Dr. James Andrews in Birmingham, Ala.
Melvin said Griffin has a sore elbow and would be examined by team physician Dr. Douglas Freedberg and have an MRI Friday in Scottsdale, Ariz.
"They're both in jeopardy of not starting the season with us," Melvin said. "Obviously, potentially it is a couple of blows for us."
Melvin said even if both are completely cleared after the examinations they would not immediately return to game action and spring training time is running short.
"We're two weeks away from the start of the season," Melvin said.
Parker, who was penciled in for the March 31 opener against Cleveland, had reconstructive elbow surgery in 2009 performed by Dr. James Andrews and missed the 2010 season.
Parker has struggled in spring training with a 10.61 ERA in three starts, giving up four runs on four hits, including two home runs, in a start Monday against the Los Angeles Dodgers. He had forearm problems at the end of last season.
"His bullpens were really good leading up to (games)," Melvin said. "We felt confident he was over the issues he had at the end of last year. But then when he started pitching in games, it would take him a while to get loose. His first outing wasn't great. The other day the first inning was rough and then he got going.
"Finally, when he was playing catch (Thursday) we were watching him and it didn't look like he was feeling great about it. I called him in and we talked and he admitted to me his forearm is a little bit tight, so we're going to be proactive and take care of that soon as we can," Melvin added.
Griffin, too, had elbow and forearm issues at the end of last season.
"It's the same area then," Melvin said. "It felt to me at the end of last year he was just running out of gas, so I'm not sure if it is the same thing or not."
Parker was 12-9 with a 3.97 ERA in 32 starts in 2013, while Griffin went 14-10 with a 3.83 ERA in 32 starts.
"That's why we have the depth we have," Melvin said. "We have Tommy Milone and Jesus Chavez has been pitching really well right now, too.
"Until we get final evaluations of what's going on with these two guys, we're very comfortable with Milone and Chavez," he continued.
Milone was 12-9 with a 4.14 ERA last season in 28 games, 26 of them starts.
Chavez made five starts last season with Triple-A Sacramento, but was used strictly in relief with Oakland in 35 appearances. Chavez, who has never started a game in the majors, is 3-0 without allowing a run in 12 2-3 innings in four exhibition games, including two starts.
"We basically had six for five with Milone," Melvin said of the rotation. "Now we have Chavez, who's been a swing guy for us and a starter in the minor leagues, a guy that we've already got stretched out to 65 pitchers, so he won't miss a beat as for as being potentially ready for the season if these two guys are not."