The Glovers' centre-half led by example to snuff out the threat of City dangerman Dean Windass and Gary Johnson's men will be disappointed to have only taken a point from this battling display.
City claimed a share of spoils when winger Muirhead took advantage of a lucky rebound to race 30 yards before letting fly with a right-foot strike from the edge of the box, but Yeovil produced the better chances throughout a hard-fought contest and should have made the long trip home with another victory under their belts.
The Glovers had a let off in the first minute when City's top scorer Windass saw his confident finish ruled out by an offside flag.
The veteran striker had been picked out by Muirhead after a Yeovil attack had broken down, but to the relief of the visiting fans the referee's assistant judged that the 36-year-old had strayed offside in the build-up.
The escape galvanised Johnson's men and the combination of Kevin Amankwaah and Kevin Gall was soon causing problems down Bradford's left flank.
Both men looked to have the beating of their markers and it was from a Lee Johnson corner on the right that Skiverton put his side ahead after seven minutes.
The centre-half could not have imagined that he would be given so much space inside the Bradford box, but with the City markers all at sea he raced onto Johnson's pin-point cross to head beyond keeper Russell Howarth from 14 yards.
With Johnson and Darren Way working hard in midfield, Bradford struggled to make any impression and City were lucky to escape when Gall's right-wing cross looked to have been blocked by Mark Bower's arm after 14 minutes.
It was almost half an hour before Weale was again called into action and this time he did superbly to keep out an angled shot from the threatening Windass.
A minute later a 25-yard snap-shot from Phil Jevons was well pushed out by Howarth and the City keeper was relieved when Argentinean striker Pablo Bastianini hit a weak shot straight at him after more good work by the former Everton player.
Just before the break Weale produced another excellent block after Muirhead raced menacingly into the Yeovil box and he was happy to see a Steven Schumacher strike fly wide of the mark a minute later.
Gall had impressed with his pace and control but the Yeovil winger was guilty of wasting two great openings just after the break.
After 52 minutes he broke clear thanks to a lucky rebound, but with players sprinting into the City box, Gall hesitated and his eventual cross was blocked by a defender.
A minute later a Way cross to far post caused more panic in the City defence but Gall couldn't bring the ball under control and the chance again went begging.
Muirhead's 64th minute equaliser was a real body blow to the Glovers but they responded well and could still have taken the three points as they finished the stronger of the two sides.
Amankwaah blazed high and wide with an angled drive and Jevons saw another strike fly wide of the mark, but the visitors were left cursing their luck in a dramatic climax.
With just two minutes to go midfielder Arron Davies let fly from the edge of the City box and after his shot rebounded off a post Johnson's follow-up strike was brilliantly saved by the agile Howarth.