However, the lead was short-lived with Junior Stanislas coolly placing a cross into the back of the net from inside the area to level the scoring in the 13th minute.
But Town were not to be undeterred and Mounié put the Terriers back in front in the 27th minute, rising highest to head home an Aaron Mooy free-kick.
Their first-half performance carried on in the second period with Mounié grabbing his second and van La Parra rounding off the scoring from the penalty spot.
Have a look at Blake Welton's quick-fire highlights from an entertaining afternoon at the John Smith's Stadium...
7 Minutes: GOAL! Mounié does brilliantly to latch onto a ball over the top before driving into the area. His cut back finds Pritchard who calmly places the ball past Begovic.
11 Minutes: Pritchard flashes a volley over the bar after meeting a flick on from Mounié however, the referee has blown for a foul in the build-up and Bournemouth have a free-kick.
14 Minutes: GOAL! Junior Stanislas equalises for the visitors, coolly placing a cross into the back of the net from inside the area.
20 Minutes: Appeals for a penalty as Scott Malone's low cross strikes a sliding Lewis Cook but the referee waves play-on.
23 Minutes: Mathias Zanka nearly puts Town back in front - he’s picked out superbly at the back post by Mooy but can’t guide his header past Begovic.
27 Minutes: GOAL! Mooy swings in a free-kick in from the left and Mounié is there to rise highest to head into the bottom corner.
31 Minutes: Jonathan Hogg does well to stop Ryan Fraser’s cross finding an unmarked Stanislas - the resultant corner from Charlie Daniels is met by Cook, but his flicked effort hits the outside of the post!
42 Minutes: Rajiv van La Parra’s cross causes problems before Pritchard’s final effort is blocked by Cook.
46 Minutes: Pritchard picks the ball up and drives into the box before slicing an effort wide.
48 Minutes: Mooy is inches wide; collecting the ball after a clever turn from Tom Ince, he drags an effort just past the far post.
50 Minutes: Bournemouth's Callum Wilson gets behind Town’s defence but Zanka does well to intercept and Jonas Lossl eventually collects.
61 Minutes: Ince heads a Mooy cross over the bar from six yards out.
67 Minutes: Goal! Mooy’s low ball into the area sees Mounié expertly squeeze it past Begovic into the bottom corner.
83 Minutes: Philip Billing picks the ball up and drives at the Bournemouth defence but with little support he tries his luck from 30 yards out - but it goes well over.
90+2 Minutes: Pritchard drives into the box and is fouled! The referee points to the spot and awards a penalty to Town.
90+3 Minutes: Mounié wants to take it but it’s van La Parra who has the honours - and he send Begovic the wrong way to make it 4-1!
Huddersfield Town (4-2-3-1): Lossl; Malone (Kongolo, 51’), Schindler, Zanka, Hadergjonaj; Mooy (Billing, 67’), Hogg; Van La Parra, Pritchard, Ince (Quaner, 65’); Mounie.
Subs Not Used: Coleman, Billing, Sabiri, Williams, Hefele.
AFC Bournemouth (4-2-3-1): Begovic; Francis, S. Cook (Smith, 70’), Ake, Daniel; Cook, Gosling; Stanislas (Mousset, 65’), Fraser, Ibe (King, 58’); Wilson.
Subs Not Used: Boruc, Surman, Arter, Simpson.
Booked: Gosling, Mousset
Half time: AFC Bournemouth 1 Huddersfield Town 2
Next match: Manchester United at the John Smith’s Stadium, FA Cup fifth round, Saturday, February 17 (kick off 5.30pm)