The result maintains the Terriers' impressive home form this season, which has seen them emerge victorious in three of their six games, while Tony Pulis' side remain without a win in the Premier League since August 19.
A poor first half was lit up in the 44th minute when Van La Parra curled home magnificently from 30 yards past Ben Foster.
David Wagner's Town side made life difficult for themselves shortly after the restart when German defender Christopher Schindler was sent off for picking up two yellow cards.
Pulis immediately made a triple substitution in a bid to capitalise on their numerical advantage, but Huddersfield's defence stood firm to hold on for a vital three points.
The Baggies have now gone nine games without a win in the Premier League and with a home clash against Chelsea waiting in a fortnight, things do not look like getting any easier for the under-fire Pulis.
Huddersfield, who handed a first Premier League start to Swiss defender Florent Hadergjonaj, started on the front foot and had the first sight of goal after 12 minutes when Van La Parra fired over from the edge of the penalty area.
Tom Ince, still waiting for his first Terriers goal, then saw his long-range effort well held by Foster as Wagner's side looked to take advantage of their early superiority.
Town could have fallen behind midway through the first half, though, when Jay Rodriguez latched on to Hal Robson-Kanu's throughball. The former Southampton striker lacked conviction, however, and his hesitance allowed Schindler to scurry across and block his goal-bound shot.
Ince's elusive first goal for Town should have come after 31 minutes when a corner fell invitingly into his path 12 yards out, but the forward blazed over under pressure from Kieran Gibbs.
The game sparked into life one minute before the break when Van La Parra received Scott Malone's short pass, moved the ball out of his feet, and whipped a stunning 30-yard strike into the top corner of Foster's net.
The euphoria created by Van La Parra's moment of magic was wiped out 11 minutes after the restart when Schindler picked up his second yellow card for a rash challenge on Ahmed Hegazi.
Pulis responded immediately by throwing on Salomon Rondon, James McClean and Matt Phillips in a bold triple substitution. Rondon almost made a decisive contribution after 74 minutes, but his header from Jonny Evans' cross edged agonisingly past Jonas Lossl's post.
Lossl then made two stunning saves in injury time to preserve a precious three points.
First, the Dane got down well to keep out McClean's header and then brilliantly tipped Phillips' fierce drive around the post.
Key Opta stats:
- Huddersfield have now won 11 points at home in the Premier League this season – only Arsenal (15), Manchester United (15) and Manchester City (13) have won more.
- Despite playing against 10 men for 33 minutes of this match, West Brom (49.7%) still held less possession than Huddersfield (50.3%).
- West Brom have lost all eight of the competitive matches that they have conceded the first goal in since May 2017.
- This is West Brom's lowest points tally after 11 games of a Premier League season (10) since 2005-06 (8), when they were relegated.