Milton Keynes, missing the cutting edge of suspended top scorer Izale McLeod, created the better chances early on and Clive Platt fired Jon-Paul McGovern's bouncing through ball wide from the edge of the box early on.
Platt went close again when he headed McGovern's searching deep cross straight at Seasiders goalkeeper Rhys Evans from 12 yards in the 19th minute.
McGovern should have put the Dons ahead midway through the first half, but he amazingly allowed Lloyd Dyer's pin-point cut-back to roll past him eight yards out and he was then crowded out by defenders.
But Blackpool began to look dangerous on the break and Parker fired wide one-one-one under late pressure from Sean O'Hanlon after racing on to David Fox's long ball in the 29th minute.
And after Aaron Wilbraham's glancing header from Dyer's free-kick flew inches wide for the Dons, Parker put the Seasiders ahead at the other end in the 31st minute following a swift counter-attack.
Goalkeeper Evans' long kick was flicked on by Adrian Forbes and Parker broke the offside trap to sprint clear one-on-one and he gave Dons keeper Lee Harper no chance with a sweet finish, via the inside of the far post with the outside of his right foot.
Fox should have doubled Blackpool's lead when he latched on to Carl Dickinson's slide-rule pass, but he dragged his 12-yard angled drive wide of the far post, before Scott Taylor somehow blazed Platt's lay-off over the bar from eight yards at the other end for the Dons four minutes before the break.
Parker skinned O'Hanlon near the by-line and crossed the ball low for the unmarked Morrell to tuck first-time into the top corner from seven yards.
The unmarked Keith Southern should have made it 3-0 moments later, but he somehow volleyed Parker's unselfish chip over the bar from eight yards, before Parker broke offside trap again himself in the 54th minute, but he fired wide left-footed under pressure from Paul Butler.
Parker fired wide twice more at the near post, before the home side rallied in the closing stages without reward.
Wilbraham headed McGovern's cross over the bar, Dyer volleyed wildly over from eight yards and O'Hanlon somehow fired over from four yards late on as the Dons' FA Cup adventure came to an abrupt end.