Your sentence starts with a parallel series of four actions ("hopped off," "took off," "ran into," and "started swimming towards") that the triathlete (or whatever) performs. But then it adds a couple of asides ("which was neatly caught by a volunteer" and "revealing his swim trunks") that would produce a visual morass in an undifferentiated, commas-only approach to internal punctuation:
He hopped off his bicycle, which was neatly caught by a volunteer, took off his bike shorts, revealing his swim trunks, ran into the water, and started swimming towards the first marker.
As you note, one way to subordinate the secondary elements is with parentheses, but that approach would leave you with two sets of them, which is a bit much for most people (not me, obviously):
He hopped off his bicycle (which was neatly caught by a volunteer), took off his bike shorts (revealing his swim trunks), ran into the water, and started swimming towards the first marker.
Using em dashes is not an appealing alternative, since using two sets of them in one sentence makes it extremely difficult for readers to figure out which dash goes with which fragment:
He hopped off his bicycle—which was neatly caught by a volunteer—took off his bike shorts—revealing his swim trunks—ran into the water, and started swimming towards the first marker.
And mixing em dashes and parentheses is too exotic for my taste, especially since it doesn't help readers identify "hopped off his bicycle" as the first of a set of four parallels:
He hopped off his bicycle—which was neatly caught by a volunteer—took off his bike shorts (revealing his swim trunks), ran into the water, and started swimming towards the first marker.
To my mind, the strongest way to distinguish the primary (and parallel) series of elements in your sentence from the sentence's secondary elements is to use a semicolon to signal the end of each of the first three parallel limbs and to use a comma to separate the main action described within that limb from the aside (if any) that also appears there:
He hopped off his bicycle, which was neatly caught by a volunteer; took off his bike shorts, revealing his swim trunks; ran into the water; and started swimming towards the first marker.
Obviously, your universe of options expands tremendously if you allow yourself to tinker with the actual wording of the sentence. But if you want to clarify the sentence without changing a word, using semicolons and commas to indicate what goes with what may be your best bet.