In the above answer, note the similar caring tone, when the Aged Parent addresses Wemmick as "dear boy" and when Magwitch addresses Pip as "dear boy". Note the parental concern.
Wemmick's father, the ''Aged Parent'' as he calls him, is firstly introduced to Pip on his first visit to the "castle", Wemmick's home in Walworth, with every typical element that is seen in a real castle. Pip sees for the first time, a very changed personality in Wemmick at his "personal capacities". Wemmick lives his life to the fullest at his home, and does not meddle up his office work with his personal life.
As he himself remarks,
"Well; it's a good thing, you know. It brushes the Newgate cobwebs away, and pleases the Aged. You wouldn't mind being at once introduce to the Aged, would you? "
Thus Pip is introduced to the Aged for the first time in the novel. His first impression about the Aged is,
There, we found, sitting by a fire, a very old man in a flannel coat: clean, cheerful, comfortable, and well cared for, but intensely deaf.
Thus, Pip soon understands how the Aged is "well cared for", "loved" and "looked after" by Wemmick.
"the fire" symbolises homelyness, warmth and care that the Aged receives by Wemmick. This is reminds the reader of the fire at Joe's forge which was symbolical of the love, care and homeliness which Pip received there and now he despises of. Also what gives happiness to the Aged is it that, Wemmick his son himself is always there beside him through all ups and downs.
When Joe arrives at London to see Pip, his "ever the best of friends" he is deeply ashamed of his clothing, manners and behaviour. This is yet another contrast of how the Aged feels and how Joe feels. At the time of their departure, the heart broken Joe remarks,
"If there's been any fault at all today, it's mine. You and me is not two figures to be together in London; nor yet anywheres else but what is private, and be known, and understood among friends. It ain't that I am proud, but that I want to be right, as you shall never see me no more in these clothes"
Thus Dickens brings out subtle sarcasm and bitterness regarding the protagonist. As an adult, when Pip retells the story of his childhood, we see his tone of repentance, sad and ashamed of not being grateful to Joe, Biddy and the forge together making the small social unit where he truly belonged to. We see the repentance of Adult pip when he narrates his story through his own words,
"But I never thought there was anything low and small in my keeping away from Joe, because I knew she would be contemptuous of him. It was but a day gone, and Joe has brought the tears into my eyes; they had soon dried, God forgive me ! soon dried.
The treatment of the Aged is also a sharp contrast to the way that Pip rejects Magwitch, his benefactor who did so much to make "his boy" a gentleman, and even risked his life to see him.
"Yes, Pip, dear boy, I've made a gentle man on you! It's me what has done it! I swore that time, sure as ever I earned a guinea, that guinea should go to you.[...] What odds dear boy? Do I tell it, fur you to feel a obligation?Not a bit.
The abhorrence in which I held the man, the dread I had of him,the repugnance in which I shrank from him, could not have been exceeded if he had been some terrible beast.
Thus, Dickens introduces minor characters such as the Aged parent, and the "old Barley" Clara's father and the way they take care of them to show the contrast between Pip and them. However at the latter part of the novel Pip realizes his faults and forms his "new self" (as you say) by trying to make the lives of Joe and Biddy better and Magwitch's death peaceful.