Newlands had led 1-0, but Slingsby battled back and raced into a strong 4-1 lead.
Martin Cappleman's men weren't done just yet and thanks to a Ben Luntley brace and goals from Jack Hakings and Lee McLaughlin they secured a 4-4 draw and took the game to penalties.
Newlands held their nerve from the spot, sealing a 4-2 win as Drew McCoubrey, Kile Fields, Jamie Gallagher and Luntley all netted their spot-kicks.
Newlands keeper Corey Towell was the man of the moment as Slingsby missed three of their pens to put Park into the draw for the next round.
West Pier stormed into the next round after hammering The Beagle 7-1.
Pier led 3-0 at half-time, Neil Thomas firing in the first two before Jamie Bradshaw took advantage of a poor goal-kick and added a third.
Rich Tolliday hit Pier's fourth after the break, but the Beagle hit back for 4-1.
Pier weren't to be denied though, with Will Jenkinson, Aidan Crawford and a second from Tolliday sealing a comfortable 7-1 win.
Boss Andy Spivey singled out stand-in goalkeeper Mark Oldroyd and youngsters Crawford and Rhys Holder for their efforts.
Itis Itis Rovers produced a superb performance to win 4-0 on the road at Tockwith and move into the next round.
Rovers led 1-0 at the break, Ryan Matson finishing superbly after getting on the end of an Ali Jones free-kick.
Man of the match Curtis Rose then sealed the win for the away side, hitting two second-half goals to make it 3-0, before Jack Ramos headed in their fourth and final goal.
Jordan Lee also impressed in the Rovers goal in a good all-round team performance.
The only scheduled game in Division One between Filey Town and Seamer was postponed due to a waterlogged pitch.
NRCFA SATURDAY CHALLENGE CUP:
Newlands 4-4 Slingsby (Newlands won 4-2 on pens), The Beagle 1-7 West Pier, Tockwith 0-4 Itis Itis Rovers
Filey Town P-P Seamer Sports