Right-sided midfielder Shaun Scales hammered in a hat-trick in this one-sided affair, while Ash Townley and Craig Rackham banged in a brace apiece, Chrissy Hannam completing the rout.
Striker Speight and Scales shared the man of the match award for Scalby.
Barker's six-goal haul led Itis Itis Rovers to an 18-0 home romp against Old Vic.
Liam Vasey and Neil Forsyth hit a hat-trick apiece, with Sean Bloom (2), Tom Thornton, Ryan Matson, Aidan McCallion and man of the match, left-back Liam Rackham, also on target for the hosts.
An Ian Laing brace secured a 3-2 home win for Ayton at home to Snainton, all five goals being scored in an action-packed opening 45 minutes.
Laing opened the scoring with his eighth goal in six games since joining the village team, tapping home from close-range to give Ayton the early advantage.
The lead was doubled in the 15th minute, Joe Bradshaw's acrobatic scissors-kick lobbing over the visiting keeper and into the net.
The away side then got back into the game with a Ryan Collings effort.
Skipper Laing struck again from close-range to restore the two-goal lead for Sean Pinder's side but Snainton refused to submit and banged in a second goal from Collings before the half-time interval.
Lewis Coultas had the best chance of the second half for Ayton, the youngster heading straight at the Snainton keeper when well-placed, but it was the away team who had the best chance to notch as they hit the bar from the penalty spot in the 75th minute, the ball cannoning off the woodwork and over.
Pinder said: "We are still learning, this is pretty much a whole new team so it will take time for everything to gel.
"The man of the match should probably be shared between Bradshaw, who started in central midfield then stepped into centre-back, Tim Hick, also in the heart of midfield, and teenager Tyler Beck, who also started in centre midfield but then went out to the right flank."
Man of the match for Snainton was Regan "hot-shot" Hewitt.
Both teams honoured a minute's silence before kick-off for the sad passing of Scarborough FC legend Dick Hewitt earlier this week.
Falsgrave Athletic powered to a 6-2 home win against Fishburn Park.
Sam Mahoney put Faslgrave ahead, but Park levelled only for man of the match, central midfielder Marcin Paulak to level.
A 40-yard free-kick from Joe Turner made it 3-1, then a Clayton Walker double effectively sealed the win for Athletic by half-time.
The visitors pulled a second goal back after the interval with a fine free-kick of their own, but Tommy Day made sure of the win with a sixth for the home side.