The 35-year-old centre-back scored with a deflected header to put the seal on a much-needed 2-0 success, while also playing a big part at the other end as the Millers kept a first-clean sheet in nine matches.
“He’s vital to us, he’s vital to the dressing room and he’s vital to what we stand for,” said assistant manager Barker.
“One thing I would say is that if Woody wasn’t on the pitch, I’m not sure the goal would have gone in. We talk all the time that you can’t bottle what Richard Wood has – aggression and desire to go and score goals.
“He brings that to us and that is part of the reason he plays.”
Wood’s effort may yet be taken off him by the dubious goals panel after the ball took a big deflection off a City defender on its way into the net.
Barker, standing in for manger Paul Warne – who remains in self-isolation following the recent Covid-19 outbreak in the Rotherham camp – admitted that even he is unsure as to how the visitors ended up doubling their lead at Ashton Gate.
“I have absolutely no idea what happened. I just saw loads of people jump up and the ball go into the back of the net.
“He claims it, he wants it, I’ll give it him if I have any option.”
Veteran Wood wrapped things up for Rotherham 17 minutes from time when his header from a corner from Dan Barlaser was deflected home.
Earlier, Michael Smith had handed the Millers a 44th-minute lead, beating onrushing City goalkeeper Daniel Bentley to Wes Harding’s right-wing cross and nodding into an unguarded net.
That effort was the 29-year-old’s 100th career goal and took his tally for the season to nine.
Barker said that he was pleased to see the ever-improving Smith reach such a milestone, but feels that there is still more to come from his centre-forward.
“I’m pleased for him to get his 100th goal,” the United No 2 continued. “We are very demanding of him. If I’m being honest, he should be on at least a dozen goals now. I don’t think there’s any reason he can’t end up with 14 or 15 at the end of the season.
“He’s not a million miles away but he’s got to keep working hard. He’s come a long way in a short period of time.”
Saturday’s victory lifts Rotherham to within three points of Championship safety, while they also boast three games in hand on 21st-placed Birmingham City.
Bristol City: Bentley; Hunt (Pearson 78), Mariappa, Kalas (Vyner 27), Sessegnon; Massengo (Bell 79), Walsh; Wells, Palmer (Lansbury 60), O’Dowda; Diedhiou (Semenyo 60). Unused substitutes: O’Leary, Nagy, Barkinson, Towler.
Rotherham United: Johansson; Ihiekwe, Wood, A MacDonald; Harding, Wing (Clarke 88), Barlaser (S MacDonald 88), Crooks, Giles (Olosunde 65); Ladapo (Jozefzoon 75), Smith (Hirst 87). Unused substitutes: Blackman, Wiles, Sadlier.
Referee: M Donohue (Greater Manchester).
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