Initially, one would be fairly upset if their own daughter betrayed them and had them arrested.  Parsons is not like that. Parsons feels that he did a bad thing in his transgression of thoughtcrime.  He feels that his imprisonment is a good thing because he believes in the Party.  He is scared for a bit of being executed, when he asks Winston, "You don't think the Party would arrest an innocent man, do you?"  Parsons feels that his imprisonment might give him a chance to be reeducated and thus, be saved.  It is for this reason that he asks Winston if he thinks they will shoot him, even though he has confessed.  

Parsons feels a sense of pride in terms of his daughter reporting him for arrest. He thinks that her actions are a direct reflection of his instruction as a parent: 

It was my little daughter,' said Parsons with a sort of doleful pride. 'She listened at the keyhole. Heard what I was saying, and nipped off to the patrols the very next day. Pretty smart for a nipper of seven, eh? I don't bear her any grudge for it. In fact I'm proud of her. It shows I brought her up in the right spirit, anyway.

Parsons sees her betrayal of him as representative of the good lessons he has taught her.  Parsons is so committed to being controlled by the Party that he constructs his imprisonment as an act of devotion, reflective of a "good parent" in the Oceanic system of being in the world.  It is in this where Parsons feels pride that his child has turned him in for imprisonment.  In the world of Big Brother, the only loyalty is to the state, supplanting even parental bonds with children.