LOL (Laugh Out Loud.. for future reference), no. I have not read the whole article, but I do understand where you are coming from. When they refer to the "braking parachute" it is easy to mistake or misinterpret the meaning to be "the parachute breaking or being physically damaged "with the spelling mistake of break instead of brake as pointed out by user8425. But that is not what the article means, and you can infer that without reading the whole article. Why? Because
Parachutes are made of fabrics right? So they tear; they don't break. So it is bad/wrong English to say that a parachute can break.
And for the technical aspect now. Since jet planes land at a very high speed, they can't just stop with physical brakes that apply friction on the wheels, so they have parachutes that are deployed from the rear of the jet plane. The parachutes slow the jets and help bring the plane to a stop. Since the parachutes are used for braking the run of the plane, they are referred to as the braking parachutes.
So since, the parachutes can't break, but only tear, and since high speed planes like the military jets use parachutes to help them come to a stop after landing, it is only logical to conclude they are talking about the braking parachutes used to stop the plane.
According to the sentence in question,
"The two crew members were unhurt in the accident, which the Defense Ministry said at the time was caused by a failure of the plane's breaking parachute."
the two crew members were unhurt in the accident which was caused by the failure/malfunction of the braking parachutes (parachutes that are supposed to help stop the plane once it has landed), but since the parachutes did not work as expected the plane probably crashed (I am just guessing as I have not read the article).