Cook, 67, who plays fiddle and guitar, said he was diagnosed four years ago with the disease, which robs sufferers of balance and causes tremors.
"For me, this has made it extremely frustrating to try and play guitar, fiddle or sing," Cook said in a video statement for The Tennessean newspaper.
"I’m not calling it quits but sometimes our bodies dictate what we have to do, and mine is telling me it’s time to take a break and heal."
Alabama, which also includes Cook's cousins Randy Owen and Teddy Gentry, broke in to the national country music scene in the 1980s and has sold some 75 million records. The band was named entertainer of the year - the top award - three times by the Country Music Association and was inducted into the Country Music Hall of Fame in 2005.
The band said it planned to continue to record as a trio and with hopes Cook will join them on stage from time to time.
Cook's announcement followed recent speculation that he had a substance abuse problem.
"That’s the part that hurts so bad, for people to think that he’s intoxicated or something," Owen said, saying that the trio had kept the diagnosis secret for years.