And with the recently-turned 33-year-old intending to play on for just two more seasons, he knows that this year is his best chance of finally breaking that duck.
"Winning in Europe is probably the only thing I haven't done in my career yet," said Parker, who has been with the Ospreys since the collapse of the Celtic Warriors in 2004.
"For me, that's my ultimate goal in club rugby. To win in Europe is the biggest thing. You are pushing yourself towards Europe.
"I've got another year left after this one but, to be honest, I'd like to finish on a high."
Finishing on a high would mean helping the Ospreys shake off their tag as the continent's nearly men following their European disappointments of recent seasons.
Since reaching the knockout stages for the first time in 2008, the Ospreys have fallen at the quarter-final hurdle for three successive seasons.
First there was defeat at Saracens, then a hammering at Munster and finally last season's heartbreak against Biarritz Olympique.
Parker believes those experiences have undoubtedly made the Ospreys stronger but he insists now is the time for the region to achieve its potential on the biggest stage of all.
"It's all about winning games," added Parker.
"The Magners League was a good ending to the season. We had a bit of an up and down season, really, especially with Europe. That's what we want to win.
"It was hard for us to lose in Biarritz but the boys picked it up in the Magners League Grand final against Leinster.
"We've got such high standards here…and the confidence is slowly growing. The guys are at their peak, they're training well and this should be a really exciting year for the Ospreys.
"This is also a crunch year for me. The main reason I play rugby is to win.
"I think it's going to be an exceptional year and, whether I'm playing or not, I want to do whatever I can to help the organization. I just want to be a part of something special."