The parallelism (and anaphora) in the first lines of "Song: To Celia" introduces the speaker's earnest voice—and the poem's wine metaphors.
Take a look at the repeated grammatical structure in the first quatrain:
Drink to me only with thine eyes,
And I will pledge with mine;
Or leave a kiss but in the cup,
And I’ll not look for wine.
These lines use a sing-song structure that harmonizes with the sing-song rhyme scheme. First, the speaker begs a favor from his beloved Celia. Then, he assures her of just how deeply he'll appreciate any sign of love from her.
Those assurances start with the same words: "And I will," "And I'll." That anaphora makes the speaker sound even more serious and intense, as if he's saying, "No, listen, really, I want nothing more than the tiniest loving gesture from you."
The repetitions here also introduce the poem's wine metaphor. In this impassioned speaker's eyes, a mere glance from Celia—let alone a kiss!—is more intoxicating than even the headiest drink. Parallelism subtly underscores that comparison, repeatedly drawing the reader's attention to the speaker's love-drunk state.
Parallelism thus sets up the tone and mood of the rest of the poem, making it clear that this speaker is deeply in love, and deeply serious about it.