Parolo superbly fired into the top corner a minute before the interval to give the hosts just reward for their first-half dominance before Biabiany sealed Parma's first victory in five matches just before the hour mark.
Atalanta initially responded well to falling behind but were deserted by luck and found Antonio Mirante in the Parma goal in fine form before Biabiany killed off any hopes of a comeback.
It was the Nerazzurri custodian who was the busier of the two stoppers in the early running, however, Andrea Consigli easily keeping out Amauri's strike before then pulling off a better save from the 32-year-old forward's powerful header.
While German Denis missed the target from distance for the visitors, Parma continued to pour forward and Ishak Belfodil only just failed to get on the end of Massimo Gobbi long-range pass.
Belfodil would atone for that miss in the 44th minute when he picked out Parolo, who drilled past a despairing Consigli and in to the top hand right corner.
Denis nearly levelled for his side in first-half injury time but the Argentina international saw his header from a corner rattle the crossbar.
The 31-year-old continued to plague Parma's defence in the air on the resumption and only a fine Mirante save from another header kept the scores level.
It would prove a crucial save as Biabiany put Roberto Donadoni's men two goals to the good in the 58th minute.
The Frenchman collected the ball from Amauri and calmly slotted into the bottom corner.
Parma nearly went 3-0 ahead a few minutes later when Belfodil burst into the box before firing at goal but Consigli produced a tremendous save, while at the other end, Luca Cigarini saw a free-kick only just miss the target.