When tea was called early due to rain, the visitors were 142 for four with opener Azhar Ali on 66 and first Test centurion Asad Shafiq on four.
After winning the toss and electing to bat the visitors made a cautious start, but only Ali had been able to build an innings of substance.
Bird struck twice in quick succession to give the home team the ascendancy, removing experienced duo Younis Khan (21) and Misbah-ul-Haq (11). Khan, who had been given out LBW on three but successfully appealed the decision, just looked to be finding touch when Bird (2-53) snuck a ball between his bat and pad to bowl him.
Misbah got an inside edge onto his pad and short leg Nic Maddinson dived forward to take a fine catch. Earlier Babar Azam (23) edged the last ball of the first session from Josh Hazlewood (1-15) to Australian captain Steven Smith in slips.
Smith, who had some rare catching errors in the first Test, went low to his right and just managed to cling onto the sharp chance. Pakistan won the toss and elected to bat on a dry pitch, and their openers were determined to see off any early movement from the Australian fast bowlers.
Although the pair played and missed on occasions, Sami Aslam and Ali were able to negotiate the seamers, and it wasn't until the introduction of spinner Nathan Lyon that the home side tasted success.
With his third ball, the spinner had Aslam caught at first slip by Smith. Aslam had batted with great caution for nine runs from 41 balls, but was surprised by some extra bounce and the ball caught the glove.
Ali survived a confident LBW appeal from the bowing of Hazlewood when he was on six, and was comprehensively beaten by the next ball, but steadied to anchor the innings.
Australia kept the same 11 which won the first Test of the three-match series in Brisbane by 39 runs, keeping faith in struggling rookie batsman Maddinson. Pakistan made one change, replacing seamer Rahat Ali with Sohail Khan.