New Zealand ace Parry had lost out in Monday's DPA Qualifier final to Cody Harris, with Weber suffering the same disappointment when he was edged out by Clinton Bridge in Tuesday's decider.
However, it was the three-time Oceanic Masters champion who emerged victorious when the pair did battle for the 16th place in this week's PDC World Series of Darts event at the Qantas Credit Union Arena.
Parry also overcame Kevin Luland, Don Whittington, Barry Jouannet and Rob Szabo for the loss of just two legs in his other knockout matches on Wednesday.
Weber - a qualifier in Sydney a year ago - was also in top form, whitewashing Stefan Cubrilovic and Jonathan Silcock and losing only one leg in his semi-final against Jonathan Silcock before being edged out in the final.
Parry's success means that he is competing in the Sydney Darts Masters for a third successive year, having caused a huge upset by defeating Michael van Gerwen in last year's first round.
Parry will this time be up against World Champion Gary Anderson, who is making his first appearance in the Sydney Darts Masters but has been a semi-finalist in all three PDC World Series of Darts events played so far in 2015.
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