Character Analysis
Father figures play a big role in the novel. Frex is one of the only fathers we actually see being a father. (Fiyero never has any in-text interaction with his kids, and Boq has very brief scenes with his near the novel's end.) So Frex's parental role stands out.
Frex dominates the lives of his daughters in many ways: Nessa is his mini-me and Elphaba has a hugely complex love/hate relationship with him, which reflects his feelings for her. Freud would have had a field day with those two. We get a lot of insight into Frex's complex feelings for Elphaba early in the novel, though he never verbalizes these feelings until the novel's end. But actions really can speak louder than words sometimes.
"Look, Fabala," he whispered (Melena hated the derivative, so he used it: it was his and Elphaba's private bond, the father-daughter pact against the world). "Look what I found in the forest. A little maplewood bird."
The child took the thing in her hands. [...] Frex steeled himself to hear the inevitable splintering, and to hold back his sigh of disappointment. But Elphaba did not bite. (1.6.3-4)
Frex can be an egotistical jerk who plays favorites, but he does seem to genuinely love both his daughters. He's just better at showing it with Nessa:
"But for the first couple of years – until Nessarose was born – you were a little beast. Only when saintly Nessarose was delivered to us, even more scarred than you, did you settle down like a normal child." (5.3.20)
We're sure every kid wants to hear that they're the cursed, demon child while their sibling is "saintly." In end, Frex is rather selfish – his treatment of his kids reflects himself and his own feelings; he doesn't show consideration for what his daughters might think or feel or need.
Aside from being "the dad," Frex is also linked to the novel's religious theme. He is a religious extremist who never accomplishes much but always tries very hard (like father, like daughter really). His religious beliefs are at the core of who Frex is and dictate his actions in life. Only someone with a lot of conviction (or guilt) would drag his family off to the Quadling Country to live a hard life as missionaries. But Frex isn't really a strong or brave person. He's just a flawed, desperate one:
Elphie glanced sideways at him, and for the first time she saw him as feckless – the kind of old man that Irji, if he survived, would grow to be. Constantly pawing at the edge of events, reacting instead of acting, mourning the past and praying for the future instead of stirring up the present. (4.3.6.17)
Elphaba never learns how to deal with her father's weaknesses, much as Frex never learns how to deal with her various complexities. In a way, these two characters never learn who the other is.