PETRUCHIO Antonio, my father, is deceased, And I have thrust myself into this maze, (1.2.55-56)
This is one of the first things out of Petruchio's mouth when he catches up with his old pal, Hortensio. Fine. Sharing this kind of news with a friend makes sense. The thing is, Petruchio mentions his dead dad an awful lot throughout the play. Is he sentimental? Does he miss him? Or, is he callous? Glad his dad is gone so he can take charge of the family trust and be the boss? If you were a director, how would you suggest an actor deliver these lines?
Act 2, Scene 1
TRANIO I see no reason but supposed Lucentio Must get a father, called 'supposed Vincentio'—
And that's a wonder. Fathers commonly Do get their children. But in this case of wooing, A child shall get a sire, if I fail not of my cunning. (2.1.431-435)
Here, Tranio muses about finding someone to pretend to be Lucentio's father, Vincentio, in order to finalize Lucentio's wedding contracts. Tranio cleverly puns on "get" (to find a fake father) and to "beget" (to sire, or to father). This is significant because it points to the way typical parent/child roles are reversed. (Parents are supposed to be in charge but the actions of rebellious and deceitful children throw such relationships into chaos.) In this case, a child is going to beget (invent) a fake father figure, the Pedant, behind his real father's back. In the last line, Tranio also suggests that a child (Lucentio) is going to "get" Bianca's father. That is, he's going to gain a father-in-law and he's going to get the better of his new father-in-law by eloping with Bianca and fooling Baptista. All of which helps him get rich (a common 16th-century definition for "get") in the process. That's a lot of work for one little word.
KATHERINE What, will you not suffer me? Nay, now I see She is your treasure, she must have a husband, I must dance barefoot on her wedding day And for your love to her lead apes in hell. Talk not to me. I will go sit and weep Till I can find occasion of revenge. (2.1.34-39)
Kate's accusation that Baptista loves Bianca the most sounds a bit childish, but it's not unfounded. Baptista does treat Bianca as a "treasure." This isn't the first time that word has been used to describe the way Baptista guards his youngest child. Hortensio sees Bianca as a "treasure" as well and accuses Baptista of being miserly with his "riches" (1.2.7).
KATHERINE Her silence flouts me, and I'll be revenged! (2.1.32)
Katherine and Bianca, like many sisters, have a tumultuous relationship. (Though, we don't know many women who have tied up their sisters and slapped them around, as Kate does in this scene.) We can't help but notice, however, that they never seem to make up or find any common ground and siblings or even women. Even at the play's very end, Katherine "scolds" her little sister (and the Widow). It's not just Kate, however, that can't play nice. None of the women get along, which is especially evident in a play where the men run around scheming together.
PETRUCHIO Signior Baptista, my business asketh haste, And every day I cannot come to woo. You knew my father well, and in him me, (2.1.121-123)
When Petruchio approaches Baptista for Kate's hand, he banks on the fact that Baptista knew his dead father. The good reputation of fathers is important to all the social climbing young men in this play. Here, however, we see that Petruchio sees his identity as being fused with that of his father, as though they are the same person.
Act 3, Scene 2
BAPTISTA Neighbours and friends, though bride and bridegroom wants For to supply the places at the table, You know there wants no junkets at the feast. To Tranio. Lucentio, you shall supply the bridegroom's place: And let Bianca take her sister's room. (3.2.253-259)
Baptista's response to the way Petruchio runs off with Kate after the wedding instead of sticking around to celebrate with friends and neighbors is interesting. Kate's presence, it seems, is negligible as Baptista implies that one daughter can easily replace another.
PETRUCHIO O Kate, content thee. Prithee, be not angry. KATHERINE I will be angry: what hast thou to do?—
Father, be quiet. He shall stay my leisure. (3.2.221-223)
This moment suggests that a daughter's relationship with her father is a good predictor of the kind of wife she will be. Kate is disrespectful to Baptista (we assume he has tried to interrupt Kate but she shuts him down before he can get a word in edgewise) and so, we can assume that she will be just as disobedient and disrespectful to her husband.
Act 4, Scene 4
BIONDELLO My master hath appointed me to go to Saint Luke's, to bid the priest be ready to come against you come with your appendix. (4.4.104-106)
When Biondello tells Lucentio to hurry to the church to elope with Bianca, he refers to Bianca as Lucentio's "appendix" or, appendage. This recalls the Genesis story of Eve being created from Adam's rib and implies that as his (future) wife, Bianca will cease to be a separate and whole person without her husband.
Act 5, Scene 1
VINCENTIO Art thou his father? MERCHANT, as Vincentio Ay, sir, so his mother says, if I may believe her. (5.1.34-36)
The Pedant's false and ironic statement that he knows he is Lucentio's father because Lucentio's mother is to be trusted (and not promiscuous) is interesting because mothers never actually appear on stage in the play. There are a few silly references to mothers (Kate makes a "your mama" joke at Petruchio's expense in Act 2, Scene 1) but it seems the only time moms get any props in this play is when someone is making a joke. What's up with that?
LUCENTIO Pardon, sweet father. Lucentio and Bianca kneel. VINCENTIO Lives my sweet son? (5.1.113-114)
In order for the play to set the world in its rightful place and restore the proper social order, children must reconcile with their parents. We're not sure why Vincentio forgives Lucentio so quickly and easily (aside from the obvious fact that he's happy to see his kid is alive and the play can't drag on forever). It does seem, however, that gesture must be made publicly in order for things to work out. Bianca follows suit in the next lines. Later, of course, Kate makes the exact same gesture toward Petruchio in Act 5, Scene 2, kneeling at his feet before a very public audience.