Theirs was unquestionably the greatest upset, trumping all others including the one Parks helped Glasgow Warriors achieve against the same Goliath at the same place almost five years earlier.
At 35 Connacht's former Scotland out-half may be in the twilight of a long career but he has now twice played a decisive role in the slaying of Toulouse at their citadel on the banks of the Garonne.
In the highly improbable event of Guy Noves forgetting before the Connacht fixture what Parks had done to his team on Glasgow's behalf five seasons ago, the longest-reigning club coach would have been tormented at the end by the nightmarish reality that lightning does strike the same place twice.
When the Warriors landed in Toulouse for the penultimate pool game in January 2009, the locals were in their not unaccustomed role of defending champions.
Parks changed all that with a barrage of seven goals in a Scottish victory which cost the holders a home quarter-final and diverted them on a losing journey to Cardiff.
Danny Boy didn't require quite as many on his return for a win which registered higher on the Richter Scale than any other since the tournament began almost 20 years ago.
The shuddering nature of its impact can be gauged from the respective positions of the teams in their domestic leagues.
Toulouse, second behind Clermont in the Top 14, had won all eight home matches, scoring 27 tries and conceding four.
Connacht, bottom of the RaboDirect PRO12, had lost eight league matches in a row, scoring eight tries and conceding 18.
The financial imbalance could hardly have been more pronounced. Toulouse, armed with an annual budget of more than €30 million and a global cast, against relatively impoverished opponents for whom the need to eke out an existence on the western rim of Europe has been a perennial struggle.
On the face of it, a no-contest. While the neutrals would have subscribed to that view, Pat Lam and his Connacht squad made the long haul from Galway to the south of France believing that theirs was far from a lost cause.
"We may be bottom of the table but we could have won every single game we played," Lam, Connacht's director of coaching, said. "We have a team with no superstars. We just try to make sure that everyone does his job."
They did that to such startling effect that their unwavering discipline ensured they won without barely allowing Toulouse into their half during the 14 minutes that remained after Thierry Dusautoir's converted try had put the four-time champions within a drop goal of doing to Connacht what the All Blacks had done to Ireland.
They could even afford to watch Parks miss two shots at goal during the finale before securing their place as giant-killers'-in-chief. This was no fluke but then Lam has previous when it comes to upsetting the odds.
When he led Northampton into the 2000 European final against Munster at Twickenham, Mick Galwey's powerful provincial squad were odds-on favourites to keep the trophy in Ireland 12 months after Ulster's win over Colomiers in Dublin.
The unfancied Saints had lost their last five Premiership matches, a slump which they attributed to the exacting demand of competing on three fronts. Winning the English Cup the previous month and going all the way to the European final stretched their season to 37 matches.
Instead, Lam's Northampton came up trumps by a single point, Paul Grayson's three penalties trumping David Wallace's try and a Jerry Holland drop goal on a day when Ronan O'Gara missed four penalties.
As John Steele, the Saints' former fly half and then director of rugby, said after the match: "I think people perceived that we would play like a tired team and maybe it took Munster aback that we were not the war-weary team many people thought."
Now that they have thrown Pool 3 wide open, Connacht will settle for nothing less than completing the mightiest of doubles when Toulouse line up at Galway on Saturday evening.
A home win followed by another against Zebre in the New Year would send Craig Clarke, John Muldoon, Michael Swift, Kieran Marmion, Parks et al into their last tie, away to Saracens, within reach of the quarter-finals.
It may be too soon to talk of the European rugby map turning green but the Irish provinces have never been more united in the winning cause. For the second round in succession, all four scored notable victories.
In the course of routing Northampton at Franklin's Gardens as they have never been routed before, Leinster served powerful notice that they intend winning the Heineken Cup for the fourth time in six seasons.
A victory which could easily have been greater than 40-7 left their vanquished opponents no option but acknowledge that they had been 'outclassed' and to apologise to their fans.
Ulster swamped Treviso to stay on course, like Leinster, for a home quarter-final. Munster's maximum-point win over Perpignan at Thomond Park raised their hopes of staying in Limerick for the knock-out stages later in the season.
For once, though, Ireland's big three provinces found themselves gloriously upstaged by their unsung western neighbour.
Fobbed off for so long as the poor relations, the so-called Cinderella Men have transformed themselves into fearless exponents of the old boxing adage, 'the bigger they are, the harder they fall'.
Connacht have done the European Cup a massive favour. The trick now will be to do it again.
Celtic giant-killings of Toulouse:
April 1, 2006:
Toulouse 35 (Tries-Y Nyanga, Y Jauzion. Cons- J-B Elissalde 2. Pens- J-B Elissalde 6. Drop-F Michalak).
Leinster 41 (Tries-B O'Driscoll, Jowitt, D Hickie, S Horgan. Cons-F Contepomi 3. Pens-F Contepomi 5).
December 16, 2006:
Toulouse 34 (Tries-C Poitrenaud 4. Cons-V Courrent 4. Pens-V Courrent 2).
Llanelli Scarlets 41 (Tries-D Daniel 2, D James, B Davies, N Thomas. Cons-S Jones 5. Pens-S Jones 2.)
January 17, 2009:
Toulouse 26 (Tries-Penalty try, V Clerc. Cons-Y Jauzion, V Clerc. Pens-J-B Elissalde 3, G du Toit).
Glasgow 33 (Tries-G Morrison, K Brown, M Evans. Cons-D Parks 3. Pens-D Parks 3. Drop-D Parks.
December 8, 2013:
Toulouse 14 (Tries-J-P Barraque, T Dusautoir. Cons-J-P Barraque, L Beauxis).
Connacht 16 (Try-K Marmion. Con-D Parks. Pens-D Parks 2. Drop-D Parks).
Team of the weekend:
15 Robbie Henshaw (Connacht)
14 Keith Earls (Munster)
13 Brian O'Driscoll (Leinster)
12 Dan Bowden (Leicester)
11 Luke Fitzgerald (Leinster)
10 Nick Evans (Harlequins)
9 Kieran Marmion (Connacht)
1 Thomas Domingo (Clermont)
2 Sean Cronin (Leinster)
3 Mike Ross (Leinster)
4 Mike McCarthy (Leinster)
5 Graham Kitchener (Leicester)
6 Rhys Ruddock (Leinster)
7 Chris Robshaw (Harlequins)
8 Jamie Heaslip (Leinster)
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