The Brazil international could soon be on his way to the Premier League, with Manchester City closing in on a deal to lure him away from Juventus.
He could, however, have arrived much earlier had Liverpool backed their judgement and pushed through an agreement with Sevilla.
In the end, they decided to spend their money elsewhere, with it difficult for then manager Gerard Houllier to justify spending £23 million a full-back.
Parry told BBC Radio Merseyside: "We had the deal - finally after enormous complications - tied up for Dani Alves and then the decision was should we buy two players or should we spend all of that on a full-back?
"Alves wasn’t the player then that he is now, but he always looked like he would be a great player.
"Would he have been a better signing than [Jermaine] Pennant and [Peter] Crouch for example? Probably yes, but it’s easy with hindsight.
“Clearly, we were trying to make that final step. We didn’t have Chelsea’s money so we were trying to be smarter. Looking back, if I could do anything different, it would be to focus on buying one at a time rather than having to buy five or six every season.
"There were a lot of great things that happened with recruitment. Gerard signed Sami Hyypia and Stephane Henchoz. My very first signing was Gary McAllister. When Gerard came to me and said he wanted him I paused for a minute and thought: ‘Really?’.
"Gerard’s rational was that he was the best striker of the ball in the Premier League and with the youngsters coming through would be great in the dressing room. What a contribution he made in that 2001 season.
"During Rafa’s era we brought in the likes of Xabi Alonso, Pepe Reina, Javier Mascherano, Luis Garcia and Fernando Torres so there were a few half decent ones."
Alves has gone on to become one of the successful players of all time, with the 23 trophies he collected with Barca complemented by five honours at Sevilla and a Serie A title and Coppa Italia success with Juventus.