The narrator is the parent (perhaps father). He is talking with his son. The kid calls himself チビちゃん. Makes sense?
so is the narrator the parent the only one actually speaking?
the statement is.
いっちゃんはうちの息子の大好きなシッターさんだ。: I think this just states that... icchan is my sons favourite baby sitter.
The first line of conversation.
「チビちゃん、どうしてもいっちゃんよりも年上になりたいんだよ、どうしてなれないの?」
chibichan, more then anything else you wish you to be older than icchichan, why can't you?
「生まれてきたときにもう際が離れてたから、しかたないんだよ」
when you were born your ages were already separated so it can't be helped.
「じゃあ、チビちゃんがにじゅうのとき、いっちゃんはよんじゅうななでもいいからおよめさんになってほしい」
well then, when chibichan turns 20, icchan will be 47, I want to her to become my wife.
「あんたがいいならいいけど、いっちゃんはいやだと思うな。。。ねえねえ、いっちゃんのどういうところが好きなの?」
you don't mind, but i don't think icchan will, hey what do you like about icchan?
「ええとね」
let me see
チビは車の中で真剣に考え始めた。
chibi inside the car started to give serious consideration.