Almost every line of the poem features some form of parallelism. Many of the line pairs share parallel syntax. Take lines 1-4, for example:
Ae fond kiss, and then we sever;
Ae fareweel, and then forever!
Deep in heart-wrung tears I'll pledge thee,
Warring sighs and groans I'll wage thee.
There is strong parallelism between lines 1-2 and then between lines 3-4. Each line is divided into two parts, and the parts parallel each other. Lines 3-4, for instance, both begin with a phrase that describes the speaker's emotions ("Deep in heart-wrung tears," "Warring sighs and groans)," pause with a caesura, and conclude with a clause, "I'll [pledge/wage] thee," that describes how he will offer up these emotions to the beloved.
The parallelism serves several functions in the poem. The two parallel lines provide essentially two statements of the same general idea, and the restatement gives further emphasis to that idea. The parallelism also develops that idea further, giving the speaker the chance to reveal new aspects of the original thought. In lines 17-18, for example, the speaker declares:
Fare thee weel, thou first and fairest!
Fare thee weel, thou best and dearest!
The first phrase of line 18 parallels the first phrase of line 17 because it repeats it exactly (making this anaphora), emphasizing the fact that the speaker is not merely uttering a conventional but goodbye but that he really, truly does hope that his beloved will be happy in the future.
The second phrase of each line describes the beloved, with line 18 further developing the thoughts of line 17. The speaker may have such strong feelings for his beloved because she is the "first" woman he has ever cared for so much and the first woman he has ever found so fair, or attractive. But those are not the only reasons why his love is so strong. She is the first woman he's cared for so much because she has the "best" or most admirable qualities. She is not only very beautiful but very "dear" to him.
Finally, the parallelism provides a high degree of regularity in the poem. The parallel structures and syntax help ensure the same number of syllables and pattern of stresses in each line. This regularity makes it easy to set the lines to a repeated musical tune, as Burns intended when he wrote the poem.