On the Standard Interpretation (let's call it) of the Third Man Argument (TMA), that argument is pointing to an infinite and vicious regress that arises from the One Over Many argument. The regress is infinite, because each time a Form is introduced to account for a collection, the argument can be reapplied to the new collection, consisting of that Form and the members of that previous collection, generating a new Form. The regress is vicious, because the Forms are precisely meant to explain the unity or commonality of the many particulars that share in them. If the One Over Many never succeeds in establishing, somewhere at the end of the line, some single, pre-eminent Form for each discernible kind, then it never succeeds in explaining how that kind is unified or somehow one. A single Form is meant to explain the unity of the kind; if the Forms for each kind are infinite, then recourse to a Form cannot explain the unity of the kind.
We have seen (What Does the Text Say?) how it is possible, and natural, to understand the conclusion of the TMA as putting forward exactly this difficulty.
These considerations give rise to various difficulties for the interpretation that Rickless has offered, that there is an additional argument implicit in the text.
1. Rickless concedes that the TMA is meant to establish an infinite regress. I ask him: Does he think the regress itself is also vicious? If so, then, I ask him: Why, on his interpretation, does Parmenides fail to point this out, but rather (as Rickless has it) presses on immediately to another difficulty, putatively based upon this one? Doesn't Parmenides show himself keen to point out every difficulty that affects the Forms? Why would he pass over this difficulty and not give it separate attention--a difficulty which is so striking, that Aristotle exploited it to great effect, and nearly all other readers of the TMA have presumed that this was the sole difficulty raised by the passage?
2. On Rickless' interpretation, we should expect to see an infinity (of some kind or other) mentioned twice in the TMA: the first infinity being the regress of Forms; the second being the multiplicity within each member of that regress. But we find only one mention of an infinity. (Readers may wish to consult once more the translation of the TMA, here.) Isn't it the simpler and more natural interpretation to identify this with the regress--and then say that the second infinity in fact is not in the passage at all?
3. Rickless' interpretation requires that an argument like the following is implicit in the TMA:
(i) There is an infinite regress of Forms.
(ii) Each Form earlier in the sequence participates in all the Forms later in the sequence.
(iii) Thus an infinite number of predicates can be asserted of each Form in the sequence.
(iv) But that of which many predicates can be asserted is itself manifold.
(v) And that of which an infinity of predicates can be asserted is itself infinitely divided.
(vi) Thus, each Form in the sequence is infinitely divided.
But it's implausible on its face to claim that all of this is implicit. Moreover, Parmenides is careful to make all of his steps explicit in every other objection that he raises: the TMA, on Rickless' interpretation, would be a wildly anomalous exception and actually a poor bit of philosophical writing.