Parkinson, who took over from Colin Ackermann midway through the 2022 season, is contracted to the club until the end of next season but Leicestershire said in a statement that he "does not currently consider himself to be in a position to commit to a contract extension".
Claude Henderson, Leicestershire's director of cricket, said: "Firstly, we want to congratulate Lewis on his new role. We have full belief in his abilities to lead Leicestershire to success moving forwards.
"We also want to thank Callum for his duties during 2022, as well as for standing firmly behind both Lewis and the club. Although we are disappointed Callum can't commit at this stage, the club will continue to have conversations with him to try and ensure his future lies with LCCC."
Hill said: "It's a really proud moment for me to be selected as captain of Leicestershire. I've loved my two seasons skippering the 50-over side, so I'm really excited for the challenge ahead.
"I want to continue the good work Callum led last year. He's still going to be an important player in our team, and I know I can draw on his experience whenever required. There are promising signs within the squad and we're all fully focused on delivering success in the County Championship next season."