Leicestershire resumed their second innings with three wickets in hand and just 41 runs in front.
An excellent half-century from Ben Mike and 29 runs from stand-in captain Parkinson took that lead to 149 before the final wicket fell.
"We showed some good character today, we scrapped hard with bat and ball but we didn't do the basics well all week," Parkinson said.
"We were in a great position first innings and didn't make it pay, then the no-ball situation in the second innings has been a problem.
"It was a really good cricket pitch, it deteriorated over the four days and I'd love to play on that every week.
"Lads have got to start holding their hands up and making themselves accountable, we're playing soft periods of cricket and it's not good enough.
"I enjoyed captaining the side, it suits my personality being quite loud and personable, I'm happy to fill in but I'd love Colin (Ackermann) to be back next week."
Glamorgan got off to a flying start in their pursuit of the victory target thanks to a brisk innings from David Lloyd but three wickets from Parkinson slowed their progress.
While Parkinson was a consistent threat the Glamorgan batters chased down the required runs in 40.4 overs with Sam Northeast undefeated on 40.
Leicestershire will be hugely disappointed to have given away such a strong position on day one and will need to work on their ill disciplined bowling that saw them bowl 25 no balls and concede 82 extras in the match.
With the wicket of Harry Swindells falling to the last ball of day three Leicestershire were in urgent need of a partnership to give them any chance of setting Glamorgan a challenging target.
That is exactly what they got from Mike and Callum Parkinson.
Mike narrowly missed out on a maiden first-class hundred in Leicestershire’s last match against Middlesex, with the final wicket falling with him stranded on 99 not out.
In this match he carried on that good form as he made a counterattacking 64 to give his team a target that was potentially defendable.
Leicestershire had looked frazzled during the last session of day three, and Glamorgan were in complete control.
That was not the case on the morning of the fourth day as Mike and Parkinson put on an eighth wicket stand worth 88.
Mike’s enterprising innings was ended in the first over with the second new ball when an attempted pull shot and got a top edge off Michael Hogan that had serious hang time.
Chris Cooke took the catch after waiting patiently underneath the skier.
It was debutant Andy Gorvin who was given the new ball at the other end and he claimed the first wicket of his career when he had Chris Wright trapped lbw for 3.
The innings was wrapped up when Hogan bowled Beuran Hendricks with Leicestershire 266 all out to set a victory target of 150.
Lloyd was in no mood to hang around when the chase got underway as he raced to 36 from just 32 balls. He was bowled attempting to sweep, the first of three Glamorgan batters to fall playing that shot off Parkinson.
Sam Northeast and Kiran Carlson shared a match defining partnership in Glamorgan’s first innings and they combined again in a stand worth 35 that got their team within touching distance of the target.
Carlson was dismissed by Scott Steel but Northeast saw his side home.
Glamorgan finished with 23 points from this match with Leicestershire claiming five bonus points.