Leicestershire were 172 behind overnight with only four second-innings wickets in hand after trailing by a daunting 318 on first innings.
They were eventually all out for 250, 68 short of forcing Derbyshire to bat again, Alex Thomson taking 3 for 50.
After Pakistan international Shan Masood’s second consecutive double century and a maiden first-class hundred by Mattie McKiernan had given Derbyshire such a handsome advantage at the halfway stage they would have seen this as a wasted opportunity had they failed to win.
Kimber picked up six boundaries and passed fifty for the fourth time in only his eighth first-class match and Parkinson battled for more than two hours but these were only a few positives that Leicestershire can take forward from a disappointing four days.
Derbyshire head coach Mickey Arthur was delighted with the victory.
He said: “I’m immensely proud of the players for the work they have put in and how we have evolved in a team in the short space of time I’ve been here.
"We’ve played 12 days of cricket so far and haven’t been outplayed on any of them.
“The boys are starting to believe in themselves and once you start believing in yourselves as a team that becomes really powerful and I couldn’t be more happy at the moment.
“Shan (Masood) has been outstanding, of course. He has led from the front and has taken a lot of batsmen with him but as good as he has been I thought Suranga Lakmal in this game has been outstanding.
"The spell he bowled today was world class and those two have galvanised the team.“We are under no illusions, we have to keep doing it and keep getting better and there will be some worse days but to win a four-day game is a tough thing, it is damn hard and they deserve to enjoy this win.”
It is only the second time in 58 years that Derbyshire have beaten their East Midlands rivals by an innings and the first time at Grace Road, as the Uptonsteel County Ground is traditionally known, since 1896.