Captain Cook made 96 and Stokes 87 to lift the hosts to 312 before the touring side, having decided not to enforce the follow-on, reached the close on 108 for no wicket, a huge lead of 362 runs with two days remaining.
Cook and Stokes came together with England reeling at 30 for four on Friday evening and they resumed on 85 for four in bright sunshine at the home of cricket.
Stokes crunched Mitchell Johnson to the extra cover boundary for his eighth four to bring up his fifty.
Cook followed him to the milestone by clipping spinner Nathan Lyon through mid-wicket, the captain needing 142 deliveries to get to his half-century in another courageous innings for his side.
Stokes, closing in on his third test hundred, chopped a ball from Mitchell Marsh on to his stumps that kept a touch low, ending a fluent fifth-wicket partnership of 145.
The powerful 24-year-old left-hander struck 13 fours and a six in another excellent innings in his burgeoning career.
England took lunch on 181 for five and Jos Buttler was out for 13 soon after the interval, nicking Lyon through to wicketkeeper Peter Nevill.
Moeen Ali played positively, lifting Lyon for a straight six, while Cook accumulated in his usual patient manner, although he was badly dropped on 63 by Steve Smith at square leg off Johnson.
The England captain, who hit 13 fours, was in sight of his 28th test century when he was undone by another skidding delivery from Marsh which he dragged back on to his stumps.
Moeen, on 39, was trapped lbw by Josh Hazlewood and the tall paceman bowled Mark Wood for four before Stuart Broad (21) edged Johnson to first slip to end the innings.
England finished 254 runs behind their opponents but at least they had a small foothold in the game following two days of complete Australian dominance.
Australia captain Michael Clarke opted to give his bowlers a rest and openers Chris Rogers and David Warner batted calmly through to stumps.
Warner, spilled by Adam Lyth at gully before he had scored, peppered the ropes with 10 fours in his unbeaten 60 and Rogers will resume on 44 not out.
Clarke must decide when to declare on Sunday before attempting to bowl England out to level the series at 1-1.
England won the first test in Cardiff by 169 runs.
(Editing by Justin Palmer)
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