A drop goal and penalty from fly-half Parks gave the Blues a 6-3 half-time advantage - scrum-half Greig Laidlaw replied with a penalty for the visitors.
The Scotland international then added another drop goal, three penalties and converted Alex Cuthbert's late try to seal the win.
Edinburgh claimed a try of their own through Lee Jones as their unbeaten start in Europe came to an end.
Despite the sides having registered wins over London Irish and Racing Metro, the opening half made for scrappy, patternless viewing, characterised by aimless tactical kicking by both sets of half-backs and stuttering Cardiff line-out display that saw several throws gratefully snaffled by Edinburgh's Argentinian lock Esteban Lozada.
Nick De Luca wastefully struck the post with a drop goal attempt from smack in front of the sticks and compatriot Parks quickly showed him how it should be done by slotting an effort of his own from just outside the visiting 22 at the other end.
Laidlaw quickly levelled matters with a 17th-minute penalty but was off target with a long-range effort 10 minutes later as the visitors, with back-rowers David Denton and Stuart McInally carrying strongly, enjoyed a good spell.
But the contest remained littered with errors, with Parks adding to the tally by missing a straightforward drop goal attempt after the hosts had finally generated a fluid passage of attacking play with Jamie Roberts and Sam Warburton leading the way.
There were a couple of bright breaks from lively scrum-half Lloyd Williams and one such forward raid, which saw him escape the clutches of several would-be tacklers, laid the platform for Parks to land a penalty right on the interval to restore Cardiff's lead.
The sparse crowd were finally given something to cheer when Gethin Jenkins, Taufa'ao Filise, Paul Tito and Warburton linked brilliantly in midfield to send the flanker gallivanting towards the Edinburgh 22.
The attack eventually came to nothing but the hosts were consoled by two Parks' penalties in quick succession which stretched their lead to a handy nine points after 51 minutes.
The Blues pack were beginning to get on top in the forward exchanges, with lock Bradley Davies getting through a ton of work, and more solid carrying yielded another Parks penalty and a lead that, given the manner of the game, looked as though it may be decisive.
It took Edinburgh, now with Laidlaw at 10 and Mike Blair at scrum-half after fly-half Harry Leonard was forced off just before half-time, until just before the hour mark to muster their first meaningful attack of the second stanza, but a forward pass from the otherwise impressive Tim Visser saw a promising position passed up.
But the Dutch winger, who qualifiers for Scotland in June, made no mistake moments later as he drew two defenders to put opposite wing Jones in at the right corner.
Laidlaw missed the testing conversion and the hosts found an instant reply when Alex Cuthbert battered his way through the defence to lay the foundations for Parks to slot his second drop goal.
And the win was sealed with six minutes remaining as replacement Scott Andrews picked off a loose pass to find Warburton, the flanker's pass fell to ground but Cuthbert gathered to cross in the corner for a converted score.
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