Published: 2020-07-19 21:51:14 BdST
West Indies, who started the day 437 runs behind England, need 43 more runs to avoid the follow-on.
Brathwaite looked well set to go on and get a century before a momentary lapse in concentration saw him caught and bowled by Ben Stokes for 75.
Brooks, who played with a steady hand at the other end to build a 76-run partnership with Brathwaite, remained unbeaten on 60 at the end of the session with Roston Chase (8 not out) for company.
Nightwatchman Alzarri Joseph was first to be dismissed earlier on Sunday when spinner Dom Bess had him caught at short leg for 32.
Shai Hope came out and stitched together a 53-run partnership with Brathwaite before falling for 25 when he was caught behind after nicking an off cutter from Sam Curran.
England had declared their first innings on day two on 469 for nine, on the back of centuries from Ben Stokes (176) and opener Dom Sibley (120).
West Indies have a 1-0 lead in the three-Test series, which is being played behind closed doors due to the COVID-19 pandemic, after winning the first Test in Southampton. The third Test will also be played in Manchester.