However, after recording a 6-3 victory, Parrott conceded his days as one of the game's leading lights are coming to an end.
"There's no way I've got five or six more seasons left in me," said Parrott, the 1991 World and UK champion, who will be 34 in May. "When I'm away I'm hankering for home, so there's no chance of me traipsing around the world when I'm 40.
"My attitude is to make the most of what I've got while it lasts. I'll know when to pack the cue away when practice becomes a chore and when competition stops giving me a buzz."
Parrott, a resident captain on BBC's A Question of Sport, is a regular guest on television shows and also spends an increasing amount of time on the after-dinner speaking circuit.
"That's the direction I want to take when I stop playing, but right now my attention is fully on snooker and trying to win trophies again," said the Liverpudlian, who has not won a title since the European Open in 1996.
Parrott stole the first frame with a yellow-to-pink clearance and moved 2-0 ahead thanks to a run of 45, before Williams seriously threatened to create a slice of tournament history.
The attacking left-hander potted the first 14 reds with 14 blacks in frame three. But, having dislodged the last red from its awkward position on the side cushion, he narrowly missed a hard pot to a baulk pocket.
Williams, disappointed to have come so close to the first 147 maximum in the 21-year history of the competition, at least became front-runner for a pounds 3,000 highest break bonus with his run of 112.
By sinking the blue from distance, plus pink and black using the rest, Parrott regained the advantage of 3-2 only for Williams to draw level with a stylish 101 break.
After that, though, Williams hit the wall and Parrott dominated the remainder of the contest. In pulling away from 3-3 to 6-3, Parrott totalled 236 points while conceding only 19. He will play the winner of the all-Irish quarter-final between Ken Doherty and Fergal O'Brien.
"Mark's supremely talented and knocks in some ridiculous pots, so the approach to playing him is very important," Parrott said. "The key is to accept that you won't get a sniff for a few frames and not to panic."