The visitors took the lead after only two minutes through a fine finish from Tommy Day in the opening attack of the match.
There were no further goals before the interval but Park came into the game more after the interval and levelled on the hour mark when sub Jack Kipling slotted home with his first touch.
The hosts went in front 10 minutes later through Marc Usher and secured their place in the semi-finals through Kane Brannick's finish after a long clearance from keeper - and player-boss - Carl Oliver was headed on in the 80th minute.
Oliver said: "It was a great performance from the lads today, to get into the semi-finals of a competition for teams in Division Two and Division Three is great for such a young team."
"William Stewart-Cross and Jack Kelly were our men of the match today."
Falsgrave chief Lee Smyth said: "We had a lot of players out injured and unavailable today, so I am proud of how well the lads kept going until the end. Centre-back Dave Pearson was our man of the match."
Eastway Sports and Edgehill 3rds will meet in the Frank White Trophy final after semi-final wins against Sherburn and Goldsborough United respectively.
Sherburn were rocked by the early dismissal of Carl Sample, the utility man being dismissed when the ball struck him on the hand from close range when he was on the goalline.
Despite his protests Sample was given his marching orders, but Eastway's Eric Hall skied the resulting penalty-kick over the bar.
The hosts did take the lead on 20 minutes when Dom Barber scored from long-range, and Sherburn's woes increased when their promising young player Sam Rackham went off injured before the interval.
Marcus Mockridge doubled Eastway's lead on the hour mark with a close-range effort, then Paul Mills' shot looked to have crossed the line but the referee ruled it was not a goal and waved play on.
Hall then wrapped up the win - and Eastway's place in the final - with a successful spot-kick.
Centre-back Jordan Dyson was named as Eastway's man of the match, while centre-back Steve Parker was star man for Sherburn, who defended stoutly despite losing the talismanic Sample so early on.
Edgehill 3rds recovered from conceding an early goal to romp to a 7-2 home win against Goldsborough in the second semi-final.
Ben Watson's clever chipped shot put the visitors in front early on, Gary Hepples levelling for the hosts direct from a corner and then the former Boro ace forced the ball home from close range just before half-time to put Edgehill in front.
After the break the home side ran riot, Hepples completing his hat-trick five minutes into the second half, a Martin Jenkinson tap-in confirming their place in the final.
Sub Danny I'Anson struck with his first touch, a rare right-footed shot, to make it 5-1 then 17-year-old centre-back Connor Sewell smashed in a sixth from the penalty spot.
Edgehill saved their best until last, Kyle O'Toole sprinting through the Goldsborough defence and smashing a fearsome shot in from a tight angle.
Watson's second proved to be little more than a consolation for a well-beaten visiting side.
Sewell was named as man of the match for a towering display at the back.