Both stories revolve around women who are victims of patriarchal and toxic masculinity. A Doll's House's Nora, however, lives a considerably better life than Cecie of The Color Purple. While Nora carries out a stagnant existence marked by the patronizing nature of her husband, Torvald, Cecie is repeatedly beaten, raped, and demeaned by multiple men before she at last is able to find her way to freedom. At the end of both stories, however, both women have decided to live free of the influence of the men in their life and find their own way by means of self-reliance.

This theme of self-reliance is more important in A Doll's House, wherein Nora abandons the chains that her family has imposed on her in order to live on her own terms. Cecie, on the other hand, only discovers herself through the other women in her life. Through the love of her more attractive sister, she learns not to blame her fellow women for her feelings of inferiority. She discovers a defiant sense of strength from Sofia after suggesting that she be beaten. She even discovers her sexuality through Shug, the blues singer. While the lens of queer theory can be applied to this relationship, it is perhaps less important than the fact that every aspect of Cecie's self, even her sexual relationship with her body, is discovered through a close-knit group of women.

While Nora finds herself distanced from an entire society that is the source of her woe, women included, Cecie's enemy is more firmly rooted in the patriarchy. The solidarity of women is one of the most important themes in The Color Purple, contrasting the more personal theme of independence in A Doll's House.

At the end of The Color Purple, the protagonist, Celie, has achieved a new level of self-awareness and self-actualization. This has taken quite a while for her to achieve, and winning the love of Shug Avery as well as realizing that she is complete with or without Shug in her life helped her substantially. A major part of Celie's empowerment has come from the knowledge that she, herself, is valuable and capable and worthy of love (from others and herself).

In A Doll's House, Nora Helmer's journey to empowerment comes much more quickly, as a result of her husband Torvald's disastrous response to learning that she took out a loan years ago, forging her father's name, so that she could pay for a trip that would save Torvald's life. Torvald accuses her of all manner of horrible things, insisting that she has no principles and is unfit to raise their children but that they must remain together for appearances. However, when the danger to Torvald's reputation is past, he returns to placating and coddling Nora, but it is too late. It is, at this point, that Nora realizes that she's always been expected to act like, and actually has acted like, a doll—first belonging to her father and then her husband.

Both Celie and Nora realize their own value as individuals as well as how they've been manipulated—Celie by Alphonso and Albert, Nora by her father and Torvald—and made to believe that their true selves are unworthy of love unless they act in accordance with what the men in their lives dictate. They both must learn to love themselves, to see the value in themselves and their actions, before they can reach the level of empowerment they both require in order to break away from the men that oppress them.

Self-realization constitutes the thematic resolution of each of these stories.

For Celie the achievement of forgiveness, emotional maturity and a realization of self and personal strength mark her character's successful journey. This arc of her story is one that essentially begins at a low point and gradually rises in terms of her well-being, state of mind, actual situation and self-regard. Her children and sister return to her at the novel's end, completing her rise from abuse and powerlessness to self-acceptance, self-empowerment and fulfillment.

Nora's journey is quite different in its dynamic. Her story follows a more traditional dramatic arc as she experiences several "ups and downs" and anticipates a specific salvation via Helmer, which does not materialize. The crux of Nora's story is quite isolated then, occurring at the moment of confrontation when Helmer discovers Nora's secret (the loan to Krogstad, the forgery and the blackmail, etc.). 

This narrative formula is distinctly different from the one that describes the growth of Celie's character. Nora exists in a more-or-less continuous state of anticipation and anxiety and, in her character's big moment, faces a crisis that forces her to change. 

She does change as a result of the crisis and the nature of this change clearly resembles the one that Celie gradually undertakes. Nora asserts herself and asserts her rights to an identity of her own.

Helmer: Before anything else, you’re a wife and mother.

Nora: I don’t believe that any more. I believe that before anything else, I’m a human being, just as much a one as you are … or at least I’m going to turn myself into one.… I want to think everything out for myself and make my own decisions.

An obvious difference between Nora's story and Celie's is that Nora leaves her children in the end where Celie is re-united with hers. Yet, the self-realization and emotional maturity that each character achieves is similar. 

Celie's transformation from a young passive girl, who is the object of violence and cruelty from her stepfather and her husband, into an independent woman with self-esteem is at the heart of The Color Purple. (eNotes)

Thematically we can say the same thing about Ibsen's play. However, we cannot say that the two stories are narratively parallel for the reasons stated above.