Such has been the Canaries’ improvement under Belgian Clement, Parkinson warned that Norwich would be a different prospect to the one Wrexham beat 3-2 at Carrow Road in September.
And the visitors made a dream start as they went ahead after just nine minutes when debutant Ali Ahmed picked out Slimane, who coolly guided his strike beyond Arthur Okonkwo from 12 yards out.
Ahmed – who joined the Norfolk outfit from Vancouver Whitecaps earlier this month – almost went from provider to scorer three minutes later as his curling strike rattled the Wrexham crossbar.
The Red Dragons settled after that rocky spell and saw Josh Windass volley Nathan Broadhead’s cross wide midway through the first half, but they levelled just a minute later.
Max Cleworth, fresh from signing a new contract with the club, crossed to Smith who guided his header into the far corner for his third goal in as many league games.
Parkinson’s men were firmly on top by this point, but Smith could only head Windass’ cross over as the game remained finely poised going into the break.
Play was halted just five minutes into the second half due to a medical emergency in the home end of the ground.
When play resumed, Norwich started on the front foot and should have regained the lead as Makama was played through on goal, only to be denied by goalkeeper Okonkwo from close range.
But Makama made no mistake three minutes later as the visitors produced a sublime counter attack that saw captain Kenny McLean thread through to Makama and his low, drilled shot flew beyond Okonkwo.
With their impressive unbeaten home record under serious threat, Parkinson sent on Matty James, Ollie Rathbone, Ryan Barnett and top scorer Kieffer Moore with just over 20 minutes left to play.
Makama and George Thomason both missed the target with efforts as the home side pushed for a late equaliser.
Wrexham appeared destined to make it 2-2 three minutes from time as Thomason crossed to Broadhead, but the Wales international’s effort was superbly kept out by Vladan Kovacevic to ensure the Canaries clinched victory.