Is there any relationship between Aquinas teaching on the analogical knowledge of God and Christ teaching in parables (ie. via analogies)?
Parables & Metaphors
To speak parabolically is to speak in metaphors, and metaphors pertain to poetry, "the least (infima) of all the sciences" because it "makes use of metaphors for the sake of (propter) representation" (Summa Theologica I q. 1 a. 9 arg./ad 1).
Unlike analogy, metaphors can be used to conceal knowledge (or reveal it):
Why speakest thou to them [the multitudes] in parables [and openly to the Apostles]? […] Because to you [the Apostles] it is given to know the mysteries of the kingdom of heaven: but to them it is not given.
And he spoke to them many things in parables. The reason is twofold[: that]
[…] by parables of this kind, sacred things would be hidden from the unbelievers, lest they blaspheme: for it was said above, “Give not that which is holy to dogs” ([Mt.] 7, 6). […]
spiritual things are hidden; for that reason, they cannot fully be made known through temporal things, hence, they have to be made known through different things; “I wish that God would speak with thee, and would open his lips to thee, that he might shew thee the secrets of wisdom” (Job 11, 5-6).
Metaphor is "improper supposition" according to John of St. Thomas. This means "that a word is predicated of something which does not fall under what the word signifies." (McInerny Logic of Analogy p. 146).
Thesis 4 of the 24 Thomistic Theses states:
Ens, quod denominatur ab esse, non univoce de Deo ac de creaturis dicitur, nec tamen prorsus aequivoce, sed analogice, analogia tum attributionis tum proportionalitis.
Being, which derives its name from existence, is not predicated univocally of God and creatures; nor yet merely equivocally, but analogically, by the analogy both of attribution and of proportionality.
Commentary: If the actuality of existence is in God a Pure Act and is in creatures an Act mixed with Potency, Being cannot be predicated of God and creatures in an identical way [univocally]: God is self-existing, creatures have their existence from God. Still, because the effect in some manner reproduces its cause, Being does not belong to God and creatures in a totally different sense [equivocally]. Being, as predicated of God and creatures is an analogous term. Its analogy is first that of attribution, since Being appertains to creatures as far as they have it from God, to whom it appertains by essence; and is secondly that of proportionality, since the actuality of existence is intrinsic to God and creatures as existing beings. [Summa Theologiae, Iª q. 13 a. 5; Contra Gentiles, lib. 1 cap. 32 et cap. 33 et cap. 34; De potentia, q. 7 a. 7]
cf. also: "Are analogical middle terms sufficient for a valid demonstration?," a distinctive feature of Thomism, and St. Thomas on the three types of predication from his De Principiis Naturæ