But a fortuitious effort from City captain Gary Jones reduced the deficit soon after the interval before James Hanson ensured the contest finished honours-even.
Parkin said: "That was some game and our lads showed lots of character to come back from two goals down.
"We spoke about the importance of getting an early breakthrough at half-time and that's what happened.
"We built from there and showed a huge amount of spirit.
"There was plenty, in all fairness, from both sets of players out there because we knew that United, despite what the table might say, are in really good form.
"We knew we'd have to battle and we knew we'd have to fight. We showed all of those qualities and got our rewards for that.
"We gave ourselves a bit of a mountain to climb, but it speaks volumes about the character the lads have got that we climbed it.
"Attitude-wise, we were spot on. There's always little things you can pick apart in matches but, to be honest, carry on showing the same desire and the results will come."
United landed the first blow of a pulsating first half when Murphy claimed his fourth of the season after just eight minutes.
Tony McMahon, finding himself in time and space on the right flank, drove an angled centre across the visitors' penalty box for theformer Motherwell winger, ghosting beyond Stephen Darby, to prod home from close range.
Maguire doubled United's lead just before half-time. Baxter, who later flashed wide from long distance, whipped a corner towards the far post and the centre-half claimed his sixth of the campaign after beating Jones and Jon McLaughlin to head home.
City equalised when Jones' deflected shot completely wrong-footed George Long before the United goalkeeper diverted the City captain's free-kick into Hanson's path as City restored parity.
Stephen McGinn drew a fine reaction save from McLaughlin after being released by Tony McMahon, Maguire glanced wide and Jamie Murphy went close as United pressed.
- Sports & Recreation
- Jamie Murphy
- Harry Maguire