Coker and Greg Smith put on 33 for the 3rd wicket but both were back in the pavilion at 75/4 with Wanganui making it through to lunch at 84/5.
At 103/7, Wanganui needed runs from the tail to make the match competitive and that came when Simon Badger and Akash Gill combined to add 38 for the 8th wicket in the afternoon session.
Gill went on to top score for Wanganui with 35 and get his side through to 178 all out.
Williams was the pick of the home side's bowlers picking up 5/43.
Wanganui coach Warren Marr was pleased with his side's batting effort given the the pitch was heavily favouring the bowlers.
"There's a lot of positives. We batted a good number of overs there, over 70 overs which was very encouraging.
"We wanted to go into the first drinks break no more than one down and we managed that pretty well.
"We just let ourselves down a bit through some decisions in the middle order."
Wanganui's innings left Taranaki with 50 overs to try and get a 1st innings result and needing just over 3.5 runs per over to do so.
Gill picked up Dean Robinson early on but Taranaki were 75/2 with 26 overs left in the day at the time of writing.
Wanganui's next match is against Wairarapa at home on November 26.
Even if we end up losing this from here, if they chase it down there's still a lot of positives to take from this.
Active Physio Wanganui vs Taranaki at Pukekura Park, New Plymouth:
Wanganui 178 (77.5 overs) Akash Gill 35, Callum Coker 32, Simon Badger 27. Taylor Williams 5/43.
Taranaki 75/2 (25.0 overs) James McDougall 26* Ryan Fleming 22* Liam Muggeridge 16. Akash Gill 1/12.
*There were 26 overs remaining in the match at the time of writing.
