Thanks to the paper linked by Jan Doggen in the comments we can say that it is a natural outcome of the shape of the parachute.
In particular, the shape, together with the near absence of wind and the fact that the craft is descending near vertically determines that pulsating motion.
In the words of the paper:
The studies of [1, 2] demonstrated that during the motion of an object with a parachute the latter's canopy changes its shape and geometric dimensions periodically - it pulsates. In what follows, the effect of this phenomenon on the dynamics of the object-parachute system (OPS) is examined.
The paper then proceeds to derive the pulsating frequencies and the other characteristics of the motion.
So to directly answer your own question
Is the parachute specifically designed to do this?
No, it is not.
What causes it to happen?
You'll find the detailed explanation in the references of the paper mentioned above (that apparently google scholar fails to find).
But to keep it simple, it is the flow of air around the canopy.
The canopy has a hole on the top to allow some air to escape in a controlled fashion, this provides stability, as mentioned on another SE.
The crucial point is that the hole is not sufficient for all the air below the canopy to escape (and for good reason, the parachute would offer no resistance, failing at its job!), the air then escapes laterally.
Under the conditions discussed in the linked paper (vertical descent at constant/terminal velocity) this results in a natural quasi-harmonic oscillatory movement.
Perhaps all parachutes of this design/scale behave this way, and I've just never noticed it before?
Given the paper, yes they do.