First-half tries from hooker Rhys Thomas and scrum-half Lloyd Williams laid the groundwork for an 18-9 half-time lead for the hosts, by which time the visitors were a man down after centre Steven Shingler saw red for a reckless spear tackle on opposite number Dafydd Hewitt.
Parks' boot eventually sealed the hosts' win with two second-half penalties but Irish refused to buckle and bravely kept themselves in the game through the boot of full-back Tom Homer, who kicked all their points to earn what could yet prove to be a valuable losing bonus point.
But the visitors, who have lost their first two pool fixtures, will rue a nightmare opening quarter that left them trailing 10-0, down to 14 and facing an uphill struggle to get back in the game.
The Blues, buoyed by their excellent win over Racing Metro in their Pool Two opener in Paris last week, had made an assured start with a powerful Gethin Jenkins paving the way for a third-minute Parks penalty.
And more Irish ill-discipline gave the hosts the platform from which to score the opening try after just six minutes.
Parks spurned the chance of another three points to go for the corner, and the gamble paid off as a well-executed driving maul from the resulting line-out was finished off by Thomas, with the fly-half adding the extras for a 10-0 advantage.
Things would have got worse for the slow-starting visitors had Casey Laulala managed to hold the final pass in a flowing Blues counter set up by the powerful running of giant wing Alex Cuthbert and the swift hands of flanker Michael Paterson.
But the task facing the Exiles soon became even greater when centre Shingler was dismissed for his ugly spear tackle on opposite number Hewitt, playing in place of the injured Jamie Roberts.
The irony of the moment would not have been lost on Blues' flanker Sam Warburton, who was playing his first home game since the furore over his World Cup sending off for a similar tackle against France.
But, with the ERC having handed down a directive on dangerous tackles prior to the start of the tournament, Shingler could have no complaints with the decision of referee Jerome Garces.
Homer registered the visitors' first points with a 26th-minute penalty, but the Blues disrupted a scrum at the other end to allow Parks to quickly cancel out that score.
The Exiles were starting to make a better fist of the game with 14 than they had with a full complement, and some slack Cardiff play allowed Richard Thorpe to force an infringement that Homer punished with his second penalty.
Homer soon added a third penalty but Cardiff responded with their second try straight from the restart.
The excellent Paterson re-gathered Parks' drop out and when the ball came back Williams surged decisively down a cavernous blindside gap to score an unconverted try in the corner right on half-time.
It took a brilliant cover tackle from Homer to prevent a third Blues try early in the second half, the full-back scything down Cuthbert after another powerful surge by the 21-year-old.
Parks and Homer traded further penalties as Irish continued to show brave resistance despite their numerical disadvantage.
And when Cardiff again conceded a penalty at scrum time just past the hour mark, Homer struck again to bring them within losing bonus-point range, although he was short with an effort from halfway shortly after.
The Blues' responded with a concerted attack after Irish wing Adam Thompstone was stripped of the ball in the tackle, and the outnumbered visiting defence were soon watching Parks slot his fourth three-pointer of the night.
But the Welsh side could not shake their battling opponents and Homer struck again with five minutes left to set up a tense finish as Irish made an unsuccessful late bid for an unlikely win, although they did at least leave with a merited bonus point.
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