Parr opened the scoring with an early header and fellow midfielder Garvan struck from the penalty spot midway through the second half.
Burnley have now collected just one point from their opening two games after coming off second-best against Dougie Freedman's men.
Palace made a bright start and were rewarded with the opening goal after just 10 minutes.
Ryan McGivern engineered the space to cross from the left flank and Parr arrived to plant a firm header into the net from six yards out.
The goal poured confidence into the Eagles and midfielder Darren Ambrose almost added a second moments later when his 20-yard strike flew narrowly over the crossbar.
Ambrose tried his luck again in the 17th minute with a shot from 20 yards that drew a smart stop from Burnley goalkeeper Lee Grant.
The visitors gradually steadied themselves, however, and almost drew level in the 20th minute when England Under-21 striker Jay Rodriguez saw his 12-yard header well saved.
In the 35th minute another chance came and was spurned by the visitors. Ambrose fouled Keith Treacy and Wade Elliott flighted in a dangerous free-kick which midfielder Chris McCann headed over from six yards.
Moments later Aleksandar Tunchev fouled Rodriguez and Ross Wallace's free-kick flew narrowly wide of the target.
On the stroke of half-time, an inswinging corner from the right flank was taken by Garvan and found Patrick McCarthy, who could only shoot wide.
Four minutes after the restart, Palace lost Ambrose to injury and replaced him with Kagisho Dikgacoi.
Yet the Eagles continued to keep Burnley at bay and almost grabbed their second goal in the 61st minute when Jermaine Easter's 12-yard strike was blocked by Andre Amougou.
Tunchev then headed wastefully wide from Garvan's free-kick before the hosts grabbed their second to secure the points in the 65th minute.
Burnley defender Ben Mee fouled Sean Scannell and Garvan stepped up to fire a left-footed penalty into the bottom corner.
Both sides made changes in the last quarter of the game but Palace were content to see out the match as Burnley struggled to make any inroads up front.
And the Clarets lost Marvin Bartley in the last minute as he was dismissed for violent conduct following a clash with Parr.