Parallels can be drawn between the two scenes if we examine the things that are taking place within the scenes. Ask yourself, who is fighting with whom, and why? What similarities in behavior can be seen? Also, similarities in Shakespeare's language choices can be seen. You may want to analyze the language for similarities in figurative language, rhetorical schemes, and any repetition. As we are limited in space, below are a couple of ideas to help get you started.
One similarity that can be found is a similarity in imagery. In Act 1, Scene 1, the moon is a recurring image. Theseus is the first to mention the moon in reference to the passing of time. Theseus and Hippolyta plan to marry at the time of the new moon as new moons mark beginnings. Hence, Theseus opens the play by commenting on how slowly "[t]his old moon wanes" and how its speed "lingers [his] desires" (I.i.4). The moon is mentioned a second time in this scene when Hermia's father, Egeus, accuses Lysander of seducing Hermia by serenading her under her window in the moonlight. Similarly, as a symbol of nighttime and lovemaking, the moon is a recurring image in Act 2, Scene 1. In this scene, we learn of Oberon and Titania's argument, and when he sees her, he greets her with, "Ill met by moonlight, proud Titania" (II.i.61). The reference to the moon is also a reference to sexuality. Since Titania has sworn not to come to Oberon's bed, he is sarcastically stating how displeased he is to see her under the moonlight.
One parallel between the two scenes is with respect to the quarrels. In the opening scene, Egeus is having a quarrel with Hermia because she refuses to marry Demetrius and has been brought before the Duke so that he can warn her of the punishment. Egeus wants Hermia to marry Demetrius and is jealous of the fact that she prefers her own choice, Lysander, over his choice for her. Similarly in Act 2, Scene 1, we learn that Titania and Oberon are quarreling because he is jealous of a changeling boy she has in her care. The boy is particularly beautiful and Oberon wants him for his own attendant. But more importantly, Oberon is jealous because he recognizes that her affection for the boy is rather erotic. As Puck describes, Titania "[c]rowns [the boy] with flowers, and makes him all her joy" (II.i.27). We know that for her this is a type of erotic affection as later on we see her treating Bottom in the exact same way. Hence, there is a parallel in the two scenes in terms of quarrels and jealousy due to erotic behavior.