Your first generation character is an orphan - without parent nor grandparent.
When the first generation character mates and has a child, the child will become the second generation. After the first generation character leaves the child in your care, the first generation character will show up as the parent to the child.
When the second generation character mates and has a child, the child will become the third generation. After the second generation character leaves the child in your care, the second generation character will show up as the parent to the child and the first generation character will show up as the grandparent to the child.
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The USA V4/V4.5 TamaTown site will never show the parent nor grandparent when you visit their houses in TamaTown. (The V3 TamaTown site shows them, and the European V4 TamaTown site shows then.)