Both sides had chances in what was an entertaining opening half, with Ephron Mason-Clark putting the visitors ahead on 22 minutes.
Wales international Moore responded in sublime fashion for the hosts with a hat-trick in the second half – his first since doing so for Barnsley in August 2018.
Tatsuhiro Sakamoto gave the Sky Blues late hope, although Coventry suffered a first league defeat of the campaign while also missing out on what would have been a club record seventh successive league triumph.
And Parkinson felt his side’s controlled display warranted all three points.
“I was pleased with the way that we approached the game. We made it difficult for Coventry and we had some brilliant chances in the first period,” he said.
“I said to the lads ‘don’t change anything apart from that we need that clinical edge’. Credit to the lads, they really stepped up.
“In fairness to the crowd, they clapped the team off at half time and you could sense that positivity coming out in the second half. The lads have responded in terrific fashion.
“I keep saying to everyone, we’re not here to stop the opposition, we’re here to play in a disciplined manner without the ball, but I’ve got great belief in these players and some of the football we played today was at a very high level.”
Victory means Wrexham have claimed seven points from the last nine on offer in the league – a run that has seen them rise to a season-best so far of 11th place.
Lampard – who ahead of kick-off heaped praise on Wrexham for their remarkable rise up the football pyramid – admitted the Welsh side’s ability to compete in the second tier is not a shock.
When asked if Wrexham’s start to the campaign has been a surprise, the former England midfielder responded: “No, because they’ve got good players, they’ve recruited, spent a lot of money, brought in some good players, a lot of Championship experience.
“I knew it would be a tough game, physical team, at it, crowd behind them tonight, it’s what I expected. I just want to congratulate them on this evening and that’s it.”