Apollodorus' Epitome 1.24:
Theseus, arriving in the realm of Haides [Hades] with Peirithous
[Pirithous], was thoroughly deceived, for Haides on the pretense of
hospitality had them sit first upon the throne of Lethe. Their bodies
grew onto it, and were held down by the serpent's coils. Now
Peirithous remained fast there for all time, but Herakles led Theseus
back up and sent him to Athens. Thence he was driven by Menestheus and
went to Lykomedes [Lycomedes], who threw him down an abyss and killed
James George Frazer's translation of the same passage renders "the throne of Lethe" rather as "the Chair of Forgetfulness", where, like in the preceding quotation, it is singular (rather than chairs). Lethe is the personification of oblivion, or forgetfulness, who dwelt in the Underworld. In Line 147 of Seneca's play Phaedra, one of the characters mentions Theseus being "hidden away in Lethean depths [Lethaeo abditum]".
Frazer supplies the following footnote to his own translation of Epitome 1.24:
As to Theseus and Pirithous in hell, and the rescue of Theseus by
Hercules [Herakles], see above, Apollod. 2.5.12 with the note. The
great painter Polygnotus painted the two heroes seated in chairs,
Theseus holding his friend's sword and his own, while Pirithous gazed
wistfully at the now useless blades, that had done such good service
in the world of light and life. See Paus. 10.29.9. No ancient author,
however, except Apollodorus in the present passage, expressly mentions
the Chair of Forgetfulness, though Horace seems to allude to it (Hor.
Carm. 4.7.27ff.), where he speaks of “the Lethaean bonds” which held
fast Pirithous, and which his faithful friend was powerless to break.
But when Apollodorus speaks of the heroes growing to their seats, he
may be following the old poet Panyasis, who said that Theseus and
Pirithous were not pinioned to their chairs, but that the rock growing
to their flesh held them as in a vice (Paus. 10.29.9). Indeed, Theseus
stuck so fast that, on being wrenched away by Hercules, he left a
piece of his person adhering to the rock, which, according to some
people, was the reason why the Athenians ever afterwards were so
remarkably spare in that part of their frame. See Suidas, s.v.
Λίσποι; Scholiast on Aristoph. Kn. 1368; compare Aulus Gellius
The section of Apollodorus' Bibliotheka (2.5.12) referenced in the footnote, which narrates the same event, gives no details about the mechanics of the imprisonment, simply saying that Theseus and Peirithous were "bound fast." I suppose that where we get the idea of more than one "chair of forgetfulness" is in Pausanias' description of the painting by Polygnotus at Delphi in the Description of Greece 10.29.9, also referenced above in Frazer's footnote:
Lower down than Odysseus are Theseus and Peirithous sitting upon
chairs. The former is holding in his hands the sword of Peirithous and
his own. Peirithous is looking at the swords, and you might conjecture
that he is angry with them for having been useless and of no help in
their daring adventures. Panyassis the poet says that Theseus and
Peirithous did not sit chained to their chairs, but that the rock grew
to their flesh and so served as chains.
(By the way, Carlos Parada's Greek Mythology Link does mention these reference sources, specifically those of Apollodorus and Pausanias, on the webpage that your quote comes from, granted that it is in super-fine print in a long list of several other references at the bottom of the page.)