The short answer is no they don't.
So far in these playoffs, there have been an average of 7.6 power plays per game, which is actually a considerable increase over the 6.6 per game there were during the regular season.
And before you ask, no that isn't all from the Penguins-Flyers bloodbath in the first round: Taking those teams out of the equation, we're still left with an average of about 7.4 per game.
Going back the last 10 years, that's been typical. Penalty calls don't drop in the postseason, and in the last few seasons, they've even gone up.
This likely has come as the league pushes officials to keep calling penalties properly, something they are ultimately rewarded for with more work later in the postseason.
(Or, as several of you have pointed out in the comments, there may simply also be more infractions being committed as the intensity of games increases.)
There's been some good work done on this question in the past.
Statistician Gabriel Desjardins wrote about this two years ago and highlighted how penalty calls drop the longer the playoffs go on (but still not well below regular season levels).
And the folks at Nucks Misconduct have a more in-depth breakdown from prior to these playoffs that's similar to the one shown above.
The most remarkable thing about the graph above is probably not the regular season and playoff numbers, but the overall decline in penalty calls even since prior to the lockout.
Even in the Dead Puck Era, we were seeing games with 8.5 power plays a game. This year, however, that number was down to just 6.6 during the regular season, which is a 20 per cent drop from 2009 and a 43 per cent drop from the penalty filled first year after the stoppage.
You can credit that trend with at least part of the drop in goal scoring over the past few years.
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