Archie has described himself as being parent-less prior to his appearance in The Dumping Ground.
The Dumping Ground
In How to Be Perfect, Archie competes with Toni, Billie, Finn and Kazima to be adopted by May-Li when Kazima finds out that the only way she can remain in the country and avoid deportation is if she is adopted. Archie, and the others, eventually lose interest.
In Sittin' In a Tree, Archie injures himself in the garden and quickly forms a bond with May-Li and they plan what to do with the garden. Archie calls it the 'May-Li-Archie Garden'. He becomes disappointed when May-Li lets Toni and Ryan help because his finger is injured. She promises that Archie can accompany her to the garden centre to choose flowers, but later takes Toni because she couldn't find Archie. He then wrecks the garden by trampling the flowers, and is told by May-Li that he is special.