Bianconeri coach Francesco Guidolin was worried about visiting the Stadio Ennio Tardini without his suspended 20-goal striker Antonio Di Natale, but Muriel allayed his fears with a coolly-taken brace in the first half.
Parma, who dropped into the bottom half of the table, were at a loss for ideas after the break and Roberto Pereyra compounded their misery with a third goal on the hour mark.
Guidolin's men are now eighth and can reach fifth place with a home victory over Lazio next weekend, should other results go their way.
Piotor Zielinski's woeful shot from Andrea Lazzari's rebounded effort put Parma at ease in the early exchanges, and they were soon punished for sitting back.
In the 12th minute a swift team move brought Muriel into a fine position to sweep Zielinski's killer pass beyond Antonio Mirante.
Udinese continued to press, with Alessandro Lucarelli forced to stick out a leg and bring Allan down in a dangerous position.
The free-kick came to nothing, though, and Jonathan Biabiany soon tried his luck with a daisy-cutter at the other end.
Parma were suddenly brighter and Danilo was lucky to get his head on the end of an Aleandro Rosi cross to nod the ball clear of danger.
With two minutes to play before the break, Udinese doubled their lead through a familiar source.
This time it fell to Dusan Basta to complete a sweeping move with a ball into Muriel's feet, the Colombian gratefully sweeping home into an empty net.
Amauri directed a wayward shot into the stands as Parma huffed and puffed for a response in the second half.
But even by the 62nd minute mark, the game was beyond Roberto Donadoni's men as Pereyra gave Udinese a dominant lead.
Muriel was heavily involved once again but this time Mirante repelled his shot, only for Pereyra to step forward and coolly convert the rebound.
That blow finally inspired something in a lethargic Parma side and Zeljko Brkic had to be at his very best to repel a fearsome diving header from Amauri.
Parma, though, knew they were beaten and began to focus on damage limitation, with Gabriel Paletta and Lucarelli combining well to close Muriel down 15 minutes from time.
And the hosts' lack of bite showed once more when Nicola Sansone poked a boot at Amauri's ball forward, only to watch it trickle harmlessly wide.