The hosts took the lead after only two minutes through Neil Thomas but it was all Park and Buglass, making his first start for the Park first team after impressing for the reserves, from that point onwards.
Buglass scored the leveller on 10 minutes and put the visitors ahead before half-time with his second, but it was his third, 25 minutes from time, which sealed the win.
This strike was the pick of the bunch, a stunning 22-yard volley that fizzed into the net.
Park chief Shaun Mancrief was delighted with his team’s efforts and Buglass in particular.
He said: “We fought for every single ball in that game, they were all brilliant but you have to make Buglass the man of the match for a superb hat-trick in his first game for us.”
Pier boss Andy Spivey said: “We were out-fought and outplayed by a much more determined team.”
Second division Filey Town Reserves claimed a shock 5-4 home win against Division One side Sleights.
Scott Sheldon, Joe Gage, man of the match Lee Cappleman, Josh Westmoreland and Jordan Powell, who hit the fifth and winning goal, were all on target for Town as they battled back from 2-0 down.
Ben Watson netted a treble for a re-shuffled Sleights team, with their man of the match Marc Kipling hitting the other.
Third division Itis Itis Rovers battled back from a goal down to win at Division Two basement club Ayton.
Matty Bywater gave the visitors the lead with a header, the villagers having the better of the opening 15 minutes, but then three goals in quick succession from Rovers, a brace from striker Sam Pickard either side of a cracking solo effort from defender Joe Hitchcock, put the hosts in charge.
Ayton keeper Shaun Dolan, who was their man of the match, kept his side in it for most of the second half, but goals in the dying moments from Jack Ramos and former Pickering Town ace Niall Gibb sealed the win for Curtis Rose’s team.
Luke Jones was man of the match for Rovers, the former Westover hot-shot was in top form in midfield, while for Ayton, as well as Dolan, right-back Jake Adams and centre-back Jamie Millar put in solid shifts for Paul Pybus’ team.
Hunmanby United also moved into the next round with a comfortable 3-0 home win against Division two title-chasers Scalby.
Adam Ditchburn smashed in a fearsome 25-yard shot to put United in front, his first strike in seven years.
Early in the second half Tommy Wilson made it 2-0 after fine work from Dan Baldry, and then 10 minutes from time the victory was sealed by sub Cameron Dobson.
The men of the match for Hunmanby were Ditchburn and Tom Micklethwaite.
Whitby Fishermen roared to a 5-1 win at Cayton Corinthians, only seven days after the village side had hammered the Fishermen 8-1 at home in the league.
Man of the match Mark Swales and Paul Tose both banged in doubles in the first half, with a clever flicked strike from Sam Russell completing the scoring for the Fishermen as they led 5-0 at the break.
Luke Foster scored a cracker 15 minutes from time for the home side but the game was well beyond them by that time.
Jonathan Read was named man of the match for Cayton for a good game in central midfield.
There was another shock result in the only Trophy Trader League Trophy clash of the day, third division Goal Sports surging to a 5-1 win at Division Two title-chasers Seamer Sports Reserves.
Kieran Friett put the visitors ahead after a fine pass from Dan Pollard and should have made it 2-0 shortly after but missed an attempted overhead kick with the goal at his mercy.
Dan Jewitt levelled for the hosts with a fine free-kick in the 25th minute and the scores remained level until half-time.
The turning point came in the 50th minute when Seamer were awarded a penalty but Joe Wright blasted his spot-kick wide.
Tyson Stubbings put Goal Sports in front and then turned provider for the third as his well-flighted corner was met with a bullet header from Dan Pickard.
Stubbings then raced onto a long through-ball from player-boss Mark Plumpton to make it 4-1 and then subs Mike Grayshan and Ed Burrows linked up for the fifth, the latter smashing a shot past the keeper after fine work from Grayshan.
Plumpton said: “That was the best performance from a side I have managed they were all superb, every single lad could not have done any more.”