Trailing by 203 on first innings, Warwickshire went into tea on 36-6, still 167 behind.
When play resumed after the rainfall, two more wickets fell, with Keith Barker caught behind by Andrew Hodd off Steve Patterson before Coad grabbed the all-important wicket of opener William porterfield, caught at slip by Tim Bresnan for 20 to leave the hosts on 54-8.
Better news was to come for the visitors when, in the 32nd over, was clean bowled by Adil Rashid for six to leave the hosts teetering on 79-6. Then bad light intervened as stumps was called.
Resuming after lunch on 4-0, the home side lost two wickets in the first over when successive balls from David Willey had Alex Mellor caught by Tim Bresnan at first slip and Jonathan Trott adjudged lbw.
Ian Bell survived the hat-trick ball but the captain collected only a single before edging a brute of a delivery from Ben Coad to Bresnan.
Rain then lopped off 21 overs but, upon the resumption, Coad, who took five wickets in the first innings, resumed his rampage as he knocked out Sam Hain’s off-stump, trapped Tim Ambrose lbw and had Rikki Clarke caught at second slip.
Earlier in the day, Adil Rashid’s half-century (65, 96 balls, ten fours) enabled Yorkshire to cement their advantage over Warwickshire on the third morning.
Yorkshire were all out for 381, securing a first-innings lead of 203, to leave themselves well-placed to push for their first victory of the season. In a single over before lunch, Warwickshire reached four without loss.
Yorkshire had resumed on 295-6 and Rashid and Andrew Hodd extended their partnership to 52 in 16 overs before Hodd (28, 44 balls) was bowled by Keith Barker.
Rashid batted with effective simplicity to reach his 37th first-class fifty before edging Rikki Clarke to Jonathan Trott at slip. Steve Patterson perished the same way and when David Willey edged Olly Hannon-Dalby behind, the home side was left that awkward over which transpired to include a glorious cover-driven four by William Porterfield off Ben Coad.