Wales prop Keiron Assiratti returned from a calf injury among five Cardiff changes to the side that narrowly beat the Dragons on Boxing Day.
Fellow prop Danny Southworth, lock George Nott, number eight Alun Lawrence and scrum-half Johan Mulder also started.
Stand-in Ospreys skipper Dewi Lake dropped to the replacement bench, with Parry and full-back Jack Walsh named as co-captains.
Hardy, who was only the specialist number nine in the Ospreys squad with injuries to Reuben Morgan-Williams and Luke Davies, went close to the opening score, but lost the ball over the line before flanker Harri Deaves was also held up.
Wales wing Giles eventually broke the deadlock with his fine individual try following a clever Hardy pass.
In their first attack of note Cardiff responded with some classy attacking play with centre Ben Thomas gliding through the Ospreys defence twice and providing the try-scoring pass to Adams.
Ospreys returned to the power game with Hardy benefitting by diving over the ruck to score.
Cardiff captain Liam Belcher was shown a yellow card for an illegal clearout on Rhys Davies in an action that visiting flanker Dan Thomas was also involved with, but he avoided being sin-binned.
Ospreys could not take advantage of the extra man and led 12-7 at half-time.
Cardiff were restored to 15 men, but Ospreys scored early in the second half after instrumental work from wing Hopkins and lock Ryan Smith allowing Parry to score.
Ospreys were without Wales prop Gareth Thomas, who is sidelined with a calf injury until the end of January, but Garyn Phillips demonstrated his worth when he popped up with a crucial defensive turnover.
Hardy crossed for the bonus-point score as he left Cardiff wing Mason Grady trailing in his wake before Hopkins completed the Ospreys scoring.
Cardiff responded with a fine second score for Adams before Faletau charged over, with their supporters wondering where this attacking intent had been for the opening 70 minutes.
Ospreys replacement Morgan Morse was shown a yellow card for a professional foul before Sebastian crossed for the losing bonus-point score.