Clearly the two relationships that this question refers to are the relationship between Louisa Gradgrind and her father, Mr. Gradgrind, and then the relationship between Sissy Jupe and her father, who never actually appears in the novel.
Louisa Gradgrind is brought up by an ineffectual mother and an overbearing father, who educates her using his utilitarian teaching philosophy with its emphasis on "facts" and nothing else. In our first meeting of Louisa, when she and Tom go to see the circus, it is interesting to see how he completely ignores any feelings or concerns that she has about her life. In response to her being "tired," her father replies:
"Say not another word," returned Mr. Gradgrind. "You are childish. I will hear no more."
His complete denial of her feelings and emotions, in spite of the "intense and searching" looks that his daughter gives him, produces the adult Louisa that is emotionally detached from life. A key chapter for you to look at is Chapter 15 when Mr. Gradgrind relays to Louisa the proposal for her hand that he has received from Mr. Bounderby. The narrator tells us something key about their relationship in this section:
From the beginning, she had sat looking at him fixedly. As he now leaned back in his chair, and bent his deep-set eyes upon her in his turn, perhaps he might have seen one wavering moment in her, when she was impelled to throw herself upon his breast, and give him the pent-up confidences of her heart. But, to see it, he must have overleaped at a bond the artificial barriers he had for many years been erecting...
Thus Mr. Gradgrind's philosophy has distanced himself emotionally from his children and created a son and a daughter who are unable to emotionally engage in the world. Fortunately, however, Mr. Gradgrind recognises this fault and begins to work to heal it with Louisa.
In contrast, it is obvious that Sissy Jupe and her father had an incredibly close relationship. It is clear that she is devoted to him, and, as Childers in Chapter Six comments, Sissy would never believe that he has fled, leaving her:
"Because those two were one. Because they were never asunder. Because, up to this time, he seemed to dote upon her."
It is out of his love for his daughter that Mr. Jupe has left her, believing she will do better without his failing skills. However reprehensible this act of desertion is, Sissy Jupe remains faithful to her father and to his memory and the belief that he will return for her throughout the novel, clearly showing her position as a child of emotion, fancy and the imagination, in sharp contrast to Louisa.
Thus it is clear that the two father-daughter relationships in this novel are used to show the conflict between facts and fancy. Louisa is brought up as a child of facts, whereas Sissy, in spite of all her efforts to study hard, remains firmly in the camp of fancy, and as a result, both of the daughters have very different experiences and futures.